**Quantum Correlations in Successive Spin Measurements**

Ali Ahanj

*Department of Physics, Khayyam Higher Education Institute (KHEI), Mashhad, and School of Physics, Institute for Research in Fundamental Science(IPM), Tehran, Iran* 

#### **1. Introduction**

196 Measurements in Quantum Mechanics

Held, C. (in preparation). Incompatibility of standard quantum completeness and quantum

Hughes, R.I.G. (1989). *The structure and interpretation of quantum mechanics.* Cambridge,

Kochen, S. & Specker, E. (1967). The problem of hidden variables in quantum mechanics.

Landsman, K. (2009). The Born rule and its interpretation. In Greenberger et al. (2009), 64-70. Mermin, N. D. (1990). Simple unified form of the major no-hidden variables theorems.

Nielsen, M.A. & Chuang, I.L. (2000). *Quantum computation and quantum information.* 

Redhead, M. (1987). *Incompleteness, nonlocality, and realism: a prolegomenon to the philosophy of* 

Van Fraassen, B. C. (1981). A modal interpretation of quantum mechanics. In E. Beltrametti

Van Fraassen, B.C. (1991). *Quantum mechanics: an empiricist view.* Oxford: Clarendon Press. van Fraassen, B.C. and Hooker, C.A. (1976). A semantic analysis of Niels Bohr's philosophy

Von Neumann, J. (1932/1955). *Mathematische Grundlagen der Quantenmechanik.* Berlin:

and B. C. van Fraassen (eds.) *Current issues in quantum logic* (Singapore: World Scientific): 229-58. Reprinted in L. Sklar (ed.). *The philosophy of physics* (New York:

of quantum theory. In W.L. Harper and C.A. Hooker (eds.): *Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, vol. III: Foundations and Philosophy of Statistical Theories in the Physical Sciences*, (Dordrecht,

Springer, 1932; english translation: *Mathematical foundations of quantum mechanics.*

mechanics.

Mass.: Harvard University Press.

*Physical Review Letters, 65*, 3373-3376.

Garland, 2000): 279-308.

Reidel) 221-241.

Cambridge: Cambridge University Press.

*quantum mechanics.* Oxford: Clarendon Press

Princeton: Princeton University Press, 1955.

*Journal of Mathematics and Mechanics, 17*, 59-87.

Quantum Mechanics (QM) represents one of the pillars of modern physics: so far a huge amount of theoretical predictions deriving from this theory have been confirmed by very accurate experimental data. No doubts can be raised on the validity of this theory. Nevertheless, even after one century since its birth, many problems related to the interpretation of this theory persist: non-local effects of entangled states, wave function reduction and the concept of measurement in QM, the transition from a microscopic probabilistic world to a macroscopic deterministic word perfectly described by classical mechanics and so on. A possible way out from these problems would be if QM represents a statistical approximation of an unknown deterministic theory, where all observables have well defined values fixed by unknown variables, the so called Hidden Variable Theories (HVT). Therefore, the debate whether QM is a complete theory and probabilities have a non-epistemic character (i.e. nature is intrinsically probabilistic) or whether it is a statistical approximation of a deterministic theory and probabilities are due to our ignorance of some parameters (i.e. they are epistemic) dates to the beginning of the theory itself.

The fundamental paper where this problem clearly emerged appeared in 1935 when Einstein, Podolsky and Rosen asked this question by considering an explicit example (Einstein et al., 1935). For this purpose, they introduced the concept of element of reality according to the following definition: if, without disturbing a system in any way , one can predict with certainty the value of a physical quantity, then there is an element of physical reality corresponding to this quantity. They formulated also the reasonable hypothesis ( consistent with special relativity) that every non-local action was forbidden. A theory is complete when it describes every element of reality. They concluded that either one or more of their premises was wrong or Quantum Mechanics was not a complete theory, in the sense that not every element of physical reality had a counterpart in the theory.

This problem led to the search of a "complete theory" by adding hidden variables to the wave function in order to implement realism. For a long time, there was a general belief among quantum physicists that quantum mechanics can not be replaced by some complete theory (HVT) due to Von Neumann's impossibility proof ( who imposed an unwarranted constraint on HVT). But in sixties we got two theorems due to J. S. Bell (Bell, 1964) (Bell, 1966) and Kochen and Specker (Kochen & Specker, 1967). These theorems showed that quantum mechanics can

Spin Measurements 3

Quantum Correlations in Successive Spin Measurements 199

contradiction (which is not a statistical one) of quantum mechanics with local realism. Greenberger, Horne and Zeilinger (GHZ)(Greenberger et al., 1990) found a way to show more immediately, without inequalities, that results of quantum mechanics are inconsistent with the assumptions of EPR. It focuses on just one event, not the statistics of many events. Their proof relies on *eight* dimensional Hilbert space, unlike the case of Bell's theorem, which is valid in *four* dimensions. Heywood and Redhead (Heywood & Redhead, 1983) have provided a direct contradiction (without inequalities) of quantum mechanics with local realism for a particular state of two spin-1 particles. Finally, Hardy (Hardy, 1992) gave a proof of non locality

like Bell's proof but does not require inequalities. This was accomplished by considering a particular experimental setup consisting of two over-lapping Mach-Zehnder interferometers, one for positrons one for electrons, arranged so that if the electron and positron each take a particular path then they will meet and annihilate one another with probability equal to 1. This arrangement is required to produce assymetric entangled state which only exhibits non locality without any use of inequality. The argument has been generalized to two spin s particles by Clifton and Niemann (Clifton & Niemann, 1992) and to N spin half particles by

Later, Hardy showed that this kind of non locality argument can be made for almost all

This proof was again simplified by Goldstein (Goldstein, 1994) and extended it to the case of bipartite systems whose constituents belong to Hilbert spaces of arbitrary dimensions. Conceptually, as well as mathematically, space and time are differently described in quantum mechanics. While time enters as an external parameter in the dynamical evolution of a system, spatial coordinates are regarded as quantum mechanical observables. Moreover, spatially separated quantum systems are associated with the tensor product structure of the Hilbert state-space of the composite system. This allows a composite quantum system to be in a state that is not separable regardless of the spatial separation of its components. We speak about *entanglement in space*. On the other hand, time in quantum mechanics is normally regarded as lacking such a structure. Because of different roles time and space play in quantum theory one could be tempted to assume that the notion of *"entanglement in time"* cannot be introduced in

In this chapter we propose and analyze a particular scenario to account for the deviations of QM from 'realism' ( defined below), which involves correlations in the outputs of successive measurements of noncommuting operators on a spin-*s* state. The correlations for successive measurements have been used previously by Popescu (Popescu, 1995) in the context of nonlocal quantum correlations, in order to analyze a class of Werner states which are entangled but do not break (bipartite) Bell-type inequality. Although local HVT can simulate the quantum correlations between the outputs of single ideal measurement on each part of the system, it fails to simulate the correlations of the second measurements on each part. Leggett and Garg have used consecutive measurements to challenge the applicability of QM to macroscopic phenomena (Leggett & Garg, 1985). While the temporal Bell inequalities, considered in refs. (Leggett & Garg, 1985) (see also (Paz & Mahler, 1993)), are for histories, we deal here with Bell-type inequalities with predetermined measurement values at different times. The temporal Bell inequalities deal with measurement of the same observable at different times, whereas we deal here with different measurements at different times. Finally there is a large literature on the problem of information of a quantum state that can be

<sup>2</sup> . So some people were trying to show a direct

<sup>2</sup> that only requires a total of *four* dimensions in Hilbert space

<sup>2</sup> particles except for maximally entangled one (Hardy, 1993).

typically photons or particles with spin <sup>1</sup>

Pagonis and Clinton (Pagonis & Clifton, 1992).

for two particles with spin <sup>1</sup>

entangled states of two spin- <sup>1</sup>

quantum physics.

not be replaced by some classes of HVT, namely local and non-contextual HVT. The most celebrated of this kind of HVT was presented by Bohm in 1952 (Bohm, 1952). Bohm, just prior to developing his HV interpretation, introduced a simplified scenario involving two spin-half particles with correlated spins, rather than two particles with correlated positions and momenta as used by EPR. The EPR-Bohm scenario has the advantage of being experimentally accessible.

In 1964 John Bell (Bell, 1964) derived an inequality ( which is a statistical result, and is called Bell's inequality BI) using locality and reality assumptions of EPR-Bohm, and showed that the singlet state of two spin-1/2 particles violates this inequality, and hence the contradiction with quantum mechanics.

Contemporary versions of the argument are based on the Clauser, Horne, Shimony and Holt (CHSH) inequality(Clauser et al., 1969), rather than the original inequality used by Bell. There is a very good reason for that. While Bell's argument applied only to the singlet state, the CHSH inequality is violated by all pure entangled states (Gisin & Peres, 1992). Early versions of CHSH inequalities involved only two observers, each one having a choice of two (mutually incompatible) experiments. The various outcomes of each experiment were lumped into two sets, arbitrarily called +1 and −1. Possible generalizations involve more than two observers, or more than two alternative experiments for each observer, or more than two distinct outcomes for each experiment. We may consider *n*-partite systems, each subject to a choice of *m v*-valued measurements. This gives a total of (*mv*)*<sup>n</sup>* experimentally accessible probabilities. The set of Bell inequalities is then the set of inequalities that bounds this region of probabilities to those accessible with a local hidden variable model. Thus for each value of *n*, *m* and *v* the set of local realistic theories is a polytopes bounded by a finite set of linear Bell inequalities. The CHSH inequalities apply to a situation (*n*, *m*, *v*)=(2, 2, 2). Gisin *et al* (Gisin & Bechmann-Pasquinucci, 1998) have found a family of Bell inequalities for the case with the number of measurements is arbitrary, i.e. (*n*, *m*, *v*)=(2, *m*, 2). Collins et al (Collins, Gisin, Linden, Massar & Popescu., 2002) and Kaszlikowski et al (Kaszlikowski et al., n.d.) have produced inequalities for arbitrarily high dimensional systems, i.e. (*n*, *m*, *v*)=(2, 2, *v*). The most complete study of Bell inequalities is for the case (*n*, *m*, *v*)=(*n*, 2, 2). *n*-particle generalizations of the CHSH inequality were first proposed by Mermin (Mermin, 1990), and Belinskii and Klyshko (Belinskii & Klyshko, 1993), and have been extended by Werner and Wolf (Werner & Wolf, 2000), and Zukowski and Brukner (Zuckowski & Brukner, 2002) to give the complete set for two dichotomic observables per site.

On the theoretical side, " violation of Bell's inequalities" had become synonymous with "non-classical correlation", i.e., entanglement. One of the first papers in which finer distinctions were made was the construction of states with the property that they satisfy all the usual assumptions leading to the Bell inequalities, but can still not be generated by a purely classical mechanism ( are not "separable" in modern terminology) (Werner, 1989). This example pointed out a gap between the obviously entangled states ( violating a Bell inequality) and the obviously non-entangled ones, which are merely classical correlated ( separable). In 1995 Popescu (Popescu, 1995) ( and later (Bennett et al., 1996)) narrowed this gap considerably by showing that after local operations and classical communication one could "distill" entanglement, leading once again to violations, even from states not violating any Bell inequality initially. To summarize this phase: it became clear that violations of Bell inequalities, while still a good indicator for the presence of non-classical correlations by no means capture all kinds of "entanglement".

Bell inequalities are statistical predictions about measurements made on two particles,

2 Will-be-set-by-IN-TECH

not be replaced by some classes of HVT, namely local and non-contextual HVT. The most celebrated of this kind of HVT was presented by Bohm in 1952 (Bohm, 1952). Bohm, just prior to developing his HV interpretation, introduced a simplified scenario involving two spin-half particles with correlated spins, rather than two particles with correlated positions and momenta as used by EPR. The EPR-Bohm scenario has the advantage of being experimentally

In 1964 John Bell (Bell, 1964) derived an inequality ( which is a statistical result, and is called Bell's inequality BI) using locality and reality assumptions of EPR-Bohm, and showed that the singlet state of two spin-1/2 particles violates this inequality, and hence the contradiction

Contemporary versions of the argument are based on the Clauser, Horne, Shimony and Holt (CHSH) inequality(Clauser et al., 1969), rather than the original inequality used by Bell. There is a very good reason for that. While Bell's argument applied only to the singlet state, the CHSH inequality is violated by all pure entangled states (Gisin & Peres, 1992). Early versions of CHSH inequalities involved only two observers, each one having a choice of two (mutually incompatible) experiments. The various outcomes of each experiment were lumped into two sets, arbitrarily called +1 and −1. Possible generalizations involve more than two observers, or more than two alternative experiments for each observer, or more than two distinct outcomes for each experiment. We may consider *n*-partite systems, each subject to a choice of *m v*-valued measurements. This gives a total of (*mv*)*<sup>n</sup>* experimentally accessible probabilities. The set of Bell inequalities is then the set of inequalities that bounds this region of probabilities to those accessible with a local hidden variable model. Thus for each value of *n*, *m* and *v* the set of local realistic theories is a polytopes bounded by a finite set of linear Bell inequalities. The CHSH inequalities apply to a situation (*n*, *m*, *v*)=(2, 2, 2). Gisin *et al* (Gisin & Bechmann-Pasquinucci, 1998) have found a family of Bell inequalities for the case with the number of measurements is arbitrary, i.e. (*n*, *m*, *v*)=(2, *m*, 2). Collins et al (Collins, Gisin, Linden, Massar & Popescu., 2002) and Kaszlikowski et al (Kaszlikowski et al., n.d.) have produced inequalities for arbitrarily high dimensional systems, i.e. (*n*, *m*, *v*)=(2, 2, *v*). The most complete study of Bell inequalities is for the case (*n*, *m*, *v*)=(*n*, 2, 2). *n*-particle generalizations of the CHSH inequality were first proposed by Mermin (Mermin, 1990), and Belinskii and Klyshko (Belinskii & Klyshko, 1993), and have been extended by Werner and Wolf (Werner & Wolf, 2000), and Zukowski and Brukner (Zuckowski & Brukner, 2002) to give

On the theoretical side, " violation of Bell's inequalities" had become synonymous with "non-classical correlation", i.e., entanglement. One of the first papers in which finer distinctions were made was the construction of states with the property that they satisfy all the usual assumptions leading to the Bell inequalities, but can still not be generated by a purely classical mechanism ( are not "separable" in modern terminology) (Werner, 1989). This example pointed out a gap between the obviously entangled states ( violating a Bell inequality) and the obviously non-entangled ones, which are merely classical correlated ( separable). In 1995 Popescu (Popescu, 1995) ( and later (Bennett et al., 1996)) narrowed this gap considerably by showing that after local operations and classical communication one could "distill" entanglement, leading once again to violations, even from states not violating any Bell inequality initially. To summarize this phase: it became clear that violations of Bell inequalities, while still a good indicator for the presence of non-classical correlations by no

Bell inequalities are statistical predictions about measurements made on two particles,

accessible.

with quantum mechanics.

the complete set for two dichotomic observables per site.

means capture all kinds of "entanglement".

typically photons or particles with spin <sup>1</sup> <sup>2</sup> . So some people were trying to show a direct contradiction (which is not a statistical one) of quantum mechanics with local realism. Greenberger, Horne and Zeilinger (GHZ)(Greenberger et al., 1990) found a way to show more immediately, without inequalities, that results of quantum mechanics are inconsistent with the assumptions of EPR. It focuses on just one event, not the statistics of many events. Their proof relies on *eight* dimensional Hilbert space, unlike the case of Bell's theorem, which is valid in *four* dimensions. Heywood and Redhead (Heywood & Redhead, 1983) have provided a direct contradiction (without inequalities) of quantum mechanics with local realism for a particular state of two spin-1 particles. Finally, Hardy (Hardy, 1992) gave a proof of non locality for two particles with spin <sup>1</sup> <sup>2</sup> that only requires a total of *four* dimensions in Hilbert space like Bell's proof but does not require inequalities. This was accomplished by considering a particular experimental setup consisting of two over-lapping Mach-Zehnder interferometers, one for positrons one for electrons, arranged so that if the electron and positron each take a particular path then they will meet and annihilate one another with probability equal to 1. This arrangement is required to produce assymetric entangled state which only exhibits non locality without any use of inequality. The argument has been generalized to two spin s particles by Clifton and Niemann (Clifton & Niemann, 1992) and to N spin half particles by Pagonis and Clinton (Pagonis & Clifton, 1992).

Later, Hardy showed that this kind of non locality argument can be made for almost all entangled states of two spin- <sup>1</sup> <sup>2</sup> particles except for maximally entangled one (Hardy, 1993). This proof was again simplified by Goldstein (Goldstein, 1994) and extended it to the case of bipartite systems whose constituents belong to Hilbert spaces of arbitrary dimensions.

Conceptually, as well as mathematically, space and time are differently described in quantum mechanics. While time enters as an external parameter in the dynamical evolution of a system, spatial coordinates are regarded as quantum mechanical observables. Moreover, spatially separated quantum systems are associated with the tensor product structure of the Hilbert state-space of the composite system. This allows a composite quantum system to be in a state that is not separable regardless of the spatial separation of its components. We speak about *entanglement in space*. On the other hand, time in quantum mechanics is normally regarded as lacking such a structure. Because of different roles time and space play in quantum theory one could be tempted to assume that the notion of *"entanglement in time"* cannot be introduced in quantum physics.

In this chapter we propose and analyze a particular scenario to account for the deviations of QM from 'realism' ( defined below), which involves correlations in the outputs of successive measurements of noncommuting operators on a spin-*s* state. The correlations for successive measurements have been used previously by Popescu (Popescu, 1995) in the context of nonlocal quantum correlations, in order to analyze a class of Werner states which are entangled but do not break (bipartite) Bell-type inequality. Although local HVT can simulate the quantum correlations between the outputs of single ideal measurement on each part of the system, it fails to simulate the correlations of the second measurements on each part. Leggett and Garg have used consecutive measurements to challenge the applicability of QM to macroscopic phenomena (Leggett & Garg, 1985). While the temporal Bell inequalities, considered in refs. (Leggett & Garg, 1985) (see also (Paz & Mahler, 1993)), are for histories, we deal here with Bell-type inequalities with predetermined measurement values at different times. The temporal Bell inequalities deal with measurement of the same observable at different times, whereas we deal here with different measurements at different times. Finally there is a large literature on the problem of information of a quantum state that can be

Spin Measurements 5

Quantum Correlations in Successive Spin Measurements 201

of each of these (2*s* + 1)*<sup>n</sup>* outcomes is the joint probability for such combinations. Note

being performed at times *t*1, *t*2, . . ., *tn* respectively (with *t*<sup>1</sup> < *t*<sup>2</sup> < ... < *tn*) do not commute, above-mentioned joint probabilities for the outcomes are well defined because each of these spin observables act on different states (Fine, 1982) (Anderson et al., 2005) (Ballentine, 1990). We emphasize that this is the joint probability for the results of *n* actual measurements and not a joint probability distribution for hypothetical simultaneous values of *n* noncommuting observables. Moreover, various sub-beams (*i.e.*, wave functions) emerging from every Stern-Gerlach apparatus (corresponding to (2*s* + 1) outcomes) in every stage of measurement are separated without any overlap or recombination between them. In other

*<sup>S</sup>* · *<sup>a</sup>*ˆ2, . . ., �

*S* · *a*ˆ*i*, measured at time *ti*, will not have any part in the regions where the SG setups,

*<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,*s*−1, . . ., �

*S* · *a*ˆ*n*, whose measurements are

*<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,*s*−*j*−1, . . .,

*S* · *a*ˆ in the QM (spin) state |*ψ*�

*S* · *a*ˆ when the hidden

(*x*), corresponding to the eigen value *s* − *j* of the

*S* commutes with the interaction Hamiltonian, if any.

*α*(*λ*)*ρψ*(*λ*)*dλ*, (1)

*αp*(*α*|*λ*)*ρψ*(*λ*)*dλ*. (2)

*p*(*αi*, *a*ˆ*k*&*αj*, *a*ˆ�) �= *p*(*αi*; *a*ˆ*k*)*p*(*αj*; *a*ˆ*l*). (3)

*<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,*s*−*j*<sup>+</sup>1, �

*<sup>S</sup>* · *<sup>a</sup>*ˆ1, �

*<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,*s*, �

*<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,−*s*, are situated. We further assume that, between two successive measurements, the

Also, throughout the string of measurements, no component (*i.e.*, sub-beam) is blocked. It is to be mentioned here that the time of each of the measurements are measured by a common

HVT assumes that in every possible state of the system, all observables have well defined (sharp) values (Redhead, 1987). On the measurement of an observable in a given state, the value possessed by the observable in that state ( and no other value) results. To gain compatibility with QM and the experiments, a set of 'hidden' variables is introduced which is denoted collectively by *λ*. For given *λ*, the values of all observables are specified as the values of appropriate real valued functions defined over the domain Λ of possible values of hidden

*<sup>S</sup>* · *<sup>a</sup>*ˆ, we denote the value of �

variables have the value *λ* by *α*(*λ*). More generally, we may require that a value of *λ* gives the probability density *p*(*α*|*λ*) over the values of *α* rather than specifying the value of *α* (stochastic HVT). We denote the probability density function for the hidden variables in the state |*ψ*� by *ρψ* (*ρψ*(*λ*)*dλ* measures the probability that the collective hidden variable lies in the range *λ* to

*S* · *a*ˆ in the state |*ψ*� is

We now analyze the consequences of SHVT for our scenario. In general, the outputs of *k*th

by *<sup>α</sup>*. Considered as a function, *<sup>α</sup>* : <sup>Λ</sup> <sup>→</sup> *IR* , we represent the value of �

�*α*� = Λ

where the integration is over Λ defined above. In the general case (SHVT)

�*α*� = Λ

that even though the spin observables �

words, the eigen wave packet *<sup>ψ</sup>s*−*j*,*ti*,*a*ˆ*<sup>i</sup>*

for measurement of the observables �

spin state does not change with time i.e. �

observable �

�

clock.

**3. Implications of HVT**

variables. For the spin observable �

*λ* + *dλ*). Then the average value of �

and *l*th experiments may be correlated so that,

obtained by measuring the same operator successively on a single system. The research in this area is elegantly summarized in (Alter & Yamamoto, n.d.). Bell-type inequality with successive measurements was first considered by (Brukner et al., 2004). They have derived CHSH-type inequality (Clauser et al., 1969) for two successive measurements on an arbitrary state of a single qubit and have shown that every such state would violate that inequality for proper choice of the measurement settings. They have shown that the quantum mechanical correlation for three successive measurements, for any single qubit input state is the product of two consecutive correlations each of which is the correlation of two consecutive measurements – a scenario quite uncommon for spatial correlations. As an application of their approach, they have used the correlations in two successive measurements to overcome the limitations in RAM of a computer to calculate a Boolean function whose input bits are supplied sequentially in time.

We consider and analyze the correlations between the outputs of successive measurements for a general spin *S* state as against the general qubit state. We show that, for *S* > <sup>1</sup> <sup>2</sup> , the quantum mechanical correlation for three successive measurements is not a product of two successive correlations, that is, the correlations in two successive measurements. We show that for *S* = <sup>1</sup> 2 , the correlation between the outputs of measurements from *n* − *k* to *n* (last *k* out of *n* successive measurements) *k* = 0, 1, . . . , *n* − 1, depend on the measurement prior to *n* − *k*, when *k* is even, while for odd *k*, these correlations are independent of the outputs of measurements prior to *n* − *k*. Further, we show that all qubit states break the Bell type inequalities corresponding to *n* successive measurements, where *n* is any finite number. Finally, we study Hardy's nonlocality arguments for the correlations between the outputs of *n* successive measurements for all *s*-spin measurements. We show that the maximum probability of success of Hardy's argument in the successive measurement is much higher than the spatial ones in a certain sense.

The chapter is organized as follows. In Section 2 we describe the basic scenario in detail. Section 3 formulates the implications of hidden variable theory (HVT) for this scenario in terms of Bell-type inequalities. Section 4 evaluates these inequalities for mixed input states of single spin-*s* system for two and three successive measurements (considering various values of *s*). Section 5 deals with *n* successive measurements on spin-1/2 system. Section 6 explains the logical structure of Hardy's argument on time locality and, we show that no time-local stochastic HVT (SHVT) can simultaneously satisfy Hardy's argument. Finally we conclude with summary and comments in Section 7. Mathematical details are relegated to Appendices A and B.

#### **2. Basic scenario**

Consider the following sequence of measurements. A quantum particle with spin *s*, prepared in the initial state *ρ*0, is sent through a cascade of Stern-Gerlach (SG) measurements for the spin components along the directions given by the unit vectors *a*ˆ1, *a*ˆ2, *a*ˆ3,..., *a*ˆ*<sup>n</sup>* (*i.e.*, measurement of observables of the form � *S*.*a*ˆ, where � *S* = (*Sx*, *Sy*, *Sz*) is the vector of spin angular momentum operators *Sx*, *Sy*, *Sz* and *a*ˆ is a unit vector from **R**3). Each measurement has 2*s* + 1 possible outcomes. For the *i*-th measurement, we denote these outcomes (eigenvalues) by *α<sup>i</sup>* ∈ {*s*,*s* − 1, . . . , <sup>−</sup>*s*}. We denote by �*αi*� the quantum mechanical (ensemble) average �� *S* · *a*ˆ*i*�, by �*αiαj*� the average �(� *<sup>S</sup>* · *<sup>a</sup>*ˆ*i*)(� *S* · *a*ˆ*j*)� etc.

Each of the (2*s* + 1)*<sup>n</sup>* possible outcomes, which one gets after performing *n* consecutive measurements, corresponds to a particular combination of the results of the measurements at previous *n* − 1 steps and the result of the measurement at the *n*-th step. The probability

of each of these (2*s* + 1)*<sup>n</sup>* outcomes is the joint probability for such combinations. Note that even though the spin observables � *<sup>S</sup>* · *<sup>a</sup>*ˆ1, � *<sup>S</sup>* · *<sup>a</sup>*ˆ2, . . ., � *S* · *a*ˆ*n*, whose measurements are being performed at times *t*1, *t*2, . . ., *tn* respectively (with *t*<sup>1</sup> < *t*<sup>2</sup> < ... < *tn*) do not commute, above-mentioned joint probabilities for the outcomes are well defined because each of these spin observables act on different states (Fine, 1982) (Anderson et al., 2005) (Ballentine, 1990). We emphasize that this is the joint probability for the results of *n* actual measurements and not a joint probability distribution for hypothetical simultaneous values of *n* noncommuting observables. Moreover, various sub-beams (*i.e.*, wave functions) emerging from every Stern-Gerlach apparatus (corresponding to (2*s* + 1) outcomes) in every stage of measurement are separated without any overlap or recombination between them. In other words, the eigen wave packet *<sup>ψ</sup>s*−*j*,*ti*,*a*ˆ*<sup>i</sup>* (*x*), corresponding to the eigen value *s* − *j* of the observable � *S* · *a*ˆ*i*, measured at time *ti*, will not have any part in the regions where the SG setups, for measurement of the observables � *<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,*s*, � *<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,*s*−1, . . ., � *<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,*s*−*j*<sup>+</sup>1, � *<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,*s*−*j*−1, . . ., � *<sup>S</sup>* · *<sup>a</sup>*ˆ*i*+1,−*s*, are situated. We further assume that, between two successive measurements, the spin state does not change with time i.e. � *S* commutes with the interaction Hamiltonian, if any. Also, throughout the string of measurements, no component (*i.e.*, sub-beam) is blocked. It is to be mentioned here that the time of each of the measurements are measured by a common clock.

#### **3. Implications of HVT**

4 Will-be-set-by-IN-TECH

obtained by measuring the same operator successively on a single system. The research in this area is elegantly summarized in (Alter & Yamamoto, n.d.). Bell-type inequality with successive measurements was first considered by (Brukner et al., 2004). They have derived CHSH-type inequality (Clauser et al., 1969) for two successive measurements on an arbitrary state of a single qubit and have shown that every such state would violate that inequality for proper choice of the measurement settings. They have shown that the quantum mechanical correlation for three successive measurements, for any single qubit input state is the product of two consecutive correlations each of which is the correlation of two consecutive measurements – a scenario quite uncommon for spatial correlations. As an application of their approach, they have used the correlations in two successive measurements to overcome the limitations in RAM of a computer to calculate a Boolean function whose input bits are supplied sequentially

We consider and analyze the correlations between the outputs of successive measurements for

mechanical correlation for three successive measurements is not a product of two successive correlations, that is, the correlations in two successive measurements. We show that for *S* = <sup>1</sup>

the correlation between the outputs of measurements from *n* − *k* to *n* (last *k* out of *n* successive measurements) *k* = 0, 1, . . . , *n* − 1, depend on the measurement prior to *n* − *k*, when *k* is even, while for odd *k*, these correlations are independent of the outputs of measurements prior to *n* − *k*. Further, we show that all qubit states break the Bell type inequalities corresponding to *n* successive measurements, where *n* is any finite number. Finally, we study Hardy's nonlocality arguments for the correlations between the outputs of *n* successive measurements for all *s*-spin measurements. We show that the maximum probability of success of Hardy's argument in the

The chapter is organized as follows. In Section 2 we describe the basic scenario in detail. Section 3 formulates the implications of hidden variable theory (HVT) for this scenario in terms of Bell-type inequalities. Section 4 evaluates these inequalities for mixed input states of single spin-*s* system for two and three successive measurements (considering various values of *s*). Section 5 deals with *n* successive measurements on spin-1/2 system. Section 6 explains the logical structure of Hardy's argument on time locality and, we show that no time-local stochastic HVT (SHVT) can simultaneously satisfy Hardy's argument. Finally we conclude with summary and comments in Section 7. Mathematical details are relegated to Appendices

Consider the following sequence of measurements. A quantum particle with spin *s*, prepared in the initial state *ρ*0, is sent through a cascade of Stern-Gerlach (SG) measurements for the spin components along the directions given by the unit vectors *a*ˆ1, *a*ˆ2, *a*ˆ3,..., *a*ˆ*<sup>n</sup>* (*i.e.*, measurement

operators *Sx*, *Sy*, *Sz* and *a*ˆ is a unit vector from **R**3). Each measurement has 2*s* + 1 possible outcomes. For the *i*-th measurement, we denote these outcomes (eigenvalues) by *α<sup>i</sup>* ∈ {*s*,*s* −

Each of the (2*s* + 1)*<sup>n</sup>* possible outcomes, which one gets after performing *n* consecutive measurements, corresponds to a particular combination of the results of the measurements at previous *n* − 1 steps and the result of the measurement at the *n*-th step. The probability

*S* = (*Sx*, *Sy*, *Sz*) is the vector of spin angular momentum

*S*.*a*ˆ, where �

1, . . . , <sup>−</sup>*s*}. We denote by �*αi*� the quantum mechanical (ensemble) average ��

<sup>2</sup> , the quantum

*S* · *a*ˆ*i*�, by �*αiαj*�

2 ,

a general spin *S* state as against the general qubit state. We show that, for *S* > <sup>1</sup>

successive measurement is much higher than the spatial ones in a certain sense.

in time.

A and B.

**2. Basic scenario**

the average �(�

of observables of the form �

*<sup>S</sup>* · *<sup>a</sup>*ˆ*i*)(�

*S* · *a*ˆ*j*)� etc.

HVT assumes that in every possible state of the system, all observables have well defined (sharp) values (Redhead, 1987). On the measurement of an observable in a given state, the value possessed by the observable in that state ( and no other value) results. To gain compatibility with QM and the experiments, a set of 'hidden' variables is introduced which is denoted collectively by *λ*. For given *λ*, the values of all observables are specified as the values of appropriate real valued functions defined over the domain Λ of possible values of hidden variables. For the spin observable � *<sup>S</sup>* · *<sup>a</sup>*ˆ, we denote the value of � *S* · *a*ˆ in the QM (spin) state |*ψ*� by *<sup>α</sup>*. Considered as a function, *<sup>α</sup>* : <sup>Λ</sup> <sup>→</sup> *IR* , we represent the value of � *S* · *a*ˆ when the hidden variables have the value *λ* by *α*(*λ*). More generally, we may require that a value of *λ* gives the probability density *p*(*α*|*λ*) over the values of *α* rather than specifying the value of *α* (stochastic HVT). We denote the probability density function for the hidden variables in the state |*ψ*� by *ρψ* (*ρψ*(*λ*)*dλ* measures the probability that the collective hidden variable lies in the range *λ* to *λ* + *dλ*). Then the average value of � *S* · *a*ˆ in the state |*ψ*� is

$$
\langle \mathfrak{a} \rangle = \int\_{\Lambda} \mathfrak{a}(\lambda) \rho\_{\Psi}(\lambda) d\lambda,\tag{1}
$$

where the integration is over Λ defined above. In the general case (SHVT)

$$
\langle \mathfrak{a} \rangle = \int\_{\Lambda} \mathfrak{a} p(\mathfrak{a} | \lambda) \rho\_{\Psi}(\lambda) d\lambda. \tag{2}
$$

We now analyze the consequences of SHVT for our scenario. In general, the outputs of *k*th and *l*th experiments may be correlated so that,

$$p(\mathfrak{a}\_{i\prime}, \mathfrak{a}\_{k} \& \mathfrak{a}\_{j\prime}, \mathfrak{a}\_{\ell}) \neq p(\mathfrak{a}\_{i\prime}; \mathfrak{a}\_{k}) p(\mathfrak{a}\_{j\prime}; \mathfrak{a}\_{l}).\tag{3}$$

Spin Measurements 7

Quantum Correlations in Successive Spin Measurements 203

*z* + *x*�

*E*(*αi*, *αj*, *αk*, *λ*) = *E*(*αi*, *λ*)*E*(*αj*, *λ*)*E*(*αk*, *λ*), we can prove Mermin-Klyshko Inequality (MKI) (Mermin, 1990), (Belinskii & Klyshko, 1993)

<sup>2</sup>*α*3� + �*α*�

This is the Svetlichny inequality (SI) (Svetlichny, 1987),(Seevinck & Svetlichny, 2002),(Collins,

For *n* successive measurements on spin *s* system, we define the MK polynomials recursively

<sup>1</sup> = *α*�

*<sup>n</sup>* are obtained from *Mn* by interchanging all primed and non-primed *α*'s. The

<sup>1</sup>*α*<sup>2</sup> + *α*1*α*�

<sup>2</sup>*α*<sup>3</sup> + *α*�

*<sup>n</sup>*) + <sup>1</sup> <sup>2</sup> *<sup>M</sup>*�

recursive relation (12) gives, for all 1 ≤ *k* ≤ *n* − 1 (Collins, Gisin, Popescu, Roberts & Scarani,

*<sup>k</sup>*) + <sup>1</sup> <sup>2</sup> *<sup>M</sup>*�

*M*<sup>1</sup> = *α*1, *M*�

<sup>2</sup> *Mn*−1(*α<sup>n</sup>* <sup>+</sup> *<sup>α</sup>*�

<sup>2</sup> *Mn*−*k*(*Mk* <sup>+</sup> *<sup>M</sup>*�

2

(*α*1*α*<sup>2</sup> + *α*�

<sup>3</sup> + *α*1*α*�


<sup>1</sup> This is obtained by using equation (4) and the similar argument as has been used in deriving equation

1*α*2*α*3�−�*α*�

*<sup>n</sup>*−1(*α<sup>n</sup>* <sup>−</sup> *<sup>α</sup>*�

*<sup>n</sup>*−*k*(*Mk* <sup>−</sup> *<sup>M</sup>*�

<sup>2</sup> − *α*� 1*α*�

<sup>1</sup>*α*2*α*<sup>3</sup> − *α*�

1*α*� 2*α*�

*<sup>n</sup>*. (16)

�| ≤ |�*MKI*�| + |�*MKI*�

1*α*� 2*α*� <sup>3</sup>�| ≤ *s*

�| ≤ 2*s*

<sup>1</sup>, (11)

*<sup>n</sup>*), (12)

*<sup>k</sup>*). (13)

<sup>2</sup>), (14)

<sup>3</sup>). (15)

�| is obtained from equation (9) by interchanging primes

3. (9)

3. (10)

*yz* − *x*� *y*� *z*� ≤ 2*s* 3,

, *z*� ∈ {−*s*, −*s* + 1, . . . ,*s* − 1,*s*}

Similarly, using the algebraic fact

for three successive measurements,


Gisin, Popescu, Roberts & Scarani, 2002).

�| ≤ *<sup>s</sup>*3, where |�*MKI*�

with non-primes in MKI. It is easily shown that

2



*Mn* <sup>=</sup> <sup>1</sup>

*Mn* <sup>=</sup> <sup>1</sup>

*<sup>M</sup>*<sup>3</sup> <sup>=</sup> *MKI* <sup>=</sup> <sup>1</sup>

*<sup>M</sup>*<sup>2</sup> <sup>=</sup> *BI* <sup>=</sup> <sup>1</sup>

2 (*α*1*α*2*α*�

where

and1

Let |�*MKI*�

as follows:

where *M*�

(6).

2002) ,(Cabello, 2002a):

In particular, we have

We now show that in HVT,

−2*s*

*x*, *y*, *z*, *x*�

<sup>3</sup> <sup>≤</sup> *xyz*� <sup>+</sup> *xy*�

, *y*�

<sup>3</sup>� + �*α*1*α*�

However, in SHVT we suppose that these correlations have a common cause represented by a stochastic hidden variable *λ* so that

$$p(\mathfrak{a}\_{i\prime}, \mathfrak{a}\_{k} \& \mathfrak{a}\_{j\prime}, \mathfrak{a}\_{\ell} | \lambda) = p(\mathfrak{a}\_{i\prime}, \mathfrak{a}\_{k} | \lambda) p(\mathfrak{a}\_{j\prime}, \mathfrak{a}\_{l} | \lambda). \tag{4}$$

As a consequence of equation (4), the probability *<sup>p</sup>*(*αi*, *<sup>a</sup>*ˆ*k*|*λ*) obtained in a measurement ( *S* · *a*ˆ*<sup>k</sup>* say) performed at time *tk* is independent of any other measurement ( *S* · *a*ˆ*<sup>l</sup>* say) made at some earlier or later time *tl*. This is called locality in time (Leggett & Garg, 1985) (Brukner et al., 2004).

One should note that for a two dimensional quantum mechanical system, one can always assign values ( deterministically or probabilistically ) to the observables with the help of a HVT. Once the measurement is done, the system will be prepared in an output state ( namely, an eigenstate of the observable), and the earlier HVT may or may not work to reproduce the values of the observables to be measured on that output state ( prepared after the first measurement). We have considered possibility of existence of a HVT for every input qubit state which can reproduce the measurement outcomes of *n* successive measurements.

Equation (4) is the crucial equation expressing the fundamental implication of SHVT to the successive measurement scenario. We now obtain the Bell type inequalities from equation (4) which can be compared with QM. Here we assume that in HVT all probabilities corresponding to outputs of measurements account for the possible changes in the values of the observable being measured, ( due to the interaction of the measuring device and the system), occurring in the previous measurements.

Now �*αiαj*� is the expectation value of obtaining the outcome *α<sup>i</sup>* in the measurement of the observable *S*.*a*ˆ*<sup>i</sup>* at time *ti* as well as the outcome *α<sup>j</sup>* in the measurement of the observable *S*.*a*ˆ*<sup>j</sup>* at later time *tj*. Due to the HVT, we must have ( dropping *a*ˆ*k*, *a*ˆ)

$$
\langle \mathfrak{a}\_{i} \mathfrak{a}\_{j} \rangle = \int \rho(\lambda) E(\mathfrak{a}\_{i}, \mathfrak{a}\_{j}, \lambda) d\lambda,\tag{5}
$$

where

$$\begin{split} E(\mathfrak{a}\_{i}, \mathfrak{a}\_{j}, \lambda) &= \sum\_{\mathfrak{a}\_{i}, \mathfrak{a}\_{j}} \mathfrak{a}\_{i} \mathfrak{a}\_{j} p(\mathfrak{a}\_{i}, \mathfrak{a}\_{j} | \lambda) = \sum\_{\mathfrak{a}\_{i}} \mathfrak{a}\_{i} p(\mathfrak{a}\_{i} | \lambda) \sum\_{\mathfrak{a}\_{j}} \mathfrak{a}\_{j} p(\mathfrak{a}\_{j} | \lambda) \\ &= E(\mathfrak{a}\_{i}, \lambda) E(\mathfrak{a}\_{j}, \lambda) \end{split} \tag{6}$$

by equation (4). Now let us consider the case of two successive measurements, with options *a*ˆ1, *a*ˆ� <sup>1</sup> and *a*ˆ2, *a*ˆ� <sup>2</sup> respectively for measuring spin components. In each run of the experiment, a random choice between {*a*ˆ1, *a*ˆ� <sup>1</sup>} and {*a*ˆ2, *a*ˆ� <sup>2</sup>} is made. Define *θ<sup>i</sup>* (*i* = 1, 1� ) to be the angle between *a*ˆ*<sup>i</sup>* and the positive *z*-axis, *θij* (*i* = 1, 1� and *j* = 2, 2� ) is the angle between *a*ˆ*<sup>j</sup>* and *a*ˆ*i*. Using condition (6) and the result (Shimony, n.d.) (Jarrett, 1984)

$$-2\mathbf{s}^2 \le \mathbf{x}y + \mathbf{x}y' + \mathbf{x}'y - \mathbf{x}'y' \le 2\mathbf{s}^2, \quad \mathbf{x}, y, \mathbf{x}', y' \in \{-\mathbf{s}, -\mathbf{s} + 1, \dots, \mathbf{s} - 1, \mathbf{s}\}, \quad \mathbf{x}, \mathbf{x}, \mathbf{y} \in \{\mathbf{s}, -\mathbf{s} + 1, \dots, \mathbf{s}\}$$

we obtain

$$- \mathfrak{L}s^2 \le E(\mathfrak{a}\_1, \mathfrak{a}\_2, \lambda) + E(\mathfrak{a}\_1, \mathfrak{a}\_{2'}' \lambda) + E(\mathfrak{a}\_1', \mathfrak{a}\_{2'} \lambda) - E(\mathfrak{a}\_1', \mathfrak{a}\_{2'}' \lambda) \le 2s^2. \tag{7}$$

Multiplying by *ρ*(*λ*)*dλ* and integrating over Λ, we get the CHSH-type inequality (Clauser et al., 1969) (involving the hidden variable *λ*) corresponding to performing two successive measurements of spin-*s* observables on a spin-*s* initial state:

$$|\langle \partial I \rangle| = \frac{1}{2} |\langle a\_1 a\_2 \rangle + \langle a\_1 a\_2' \rangle + \langle a\_1' a\_2 \rangle - \langle a\_1' a\_2' \rangle| \le s^2. \tag{8}$$

Similarly, using the algebraic fact

$$-2s^3 \le \varkappa yz' + \varkappa y'z + \varkappa' yz - \varkappa' y'z' \le 2s^3 \varkappa$$

where

6 Will-be-set-by-IN-TECH

However, in SHVT we suppose that these correlations have a common cause represented by

As a consequence of equation (4), the probability *<sup>p</sup>*(*αi*, *<sup>a</sup>*ˆ*k*|*λ*) obtained in a measurement (

earlier or later time *tl*. This is called locality in time (Leggett & Garg, 1985) (Brukner et al.,

One should note that for a two dimensional quantum mechanical system, one can always assign values ( deterministically or probabilistically ) to the observables with the help of a HVT. Once the measurement is done, the system will be prepared in an output state ( namely, an eigenstate of the observable), and the earlier HVT may or may not work to reproduce the values of the observables to be measured on that output state ( prepared after the first measurement). We have considered possibility of existence of a HVT for every input qubit

Now �*αiαj*� is the expectation value of obtaining the outcome *α<sup>i</sup>* in the measurement of the

*S*.*a*ˆ*<sup>i</sup>* at time *ti* as well as the outcome *α<sup>j</sup>* in the measurement of the observable

*ρ*(*λ*)*E*(*αi*, *αj*, *λ*)*dλ*, (5)

*α<sup>j</sup> p*(*αj*|*λ*)

, *y*� ∈ {−*s*, −*s* + 1, . . . ,*s* − 1,*s*},

<sup>1</sup>, *α*�

2, *λ*) ≤ 2*s*

*<sup>α</sup><sup>i</sup> <sup>p</sup>*(*αi*|*λ*)∑*<sup>α</sup><sup>j</sup>*

= *E*(*αi*, *λ*)*E*(*αj*, *λ*) (6)

<sup>2</sup>} is made. Define *θ<sup>i</sup>* (*i* = 1, 1�

<sup>1</sup>, *α*2, *λ*) − *E*(*α*�

1*α*2�−�*α*�

1*α*� <sup>2</sup>�| ≤ *s*

state which can reproduce the measurement outcomes of *n* successive measurements. Equation (4) is the crucial equation expressing the fundamental implication of SHVT to the successive measurement scenario. We now obtain the Bell type inequalities from equation (4) which can be compared with QM. Here we assume that in HVT all probabilities corresponding to outputs of measurements account for the possible changes in the values of the observable being measured, ( due to the interaction of the measuring device and the system), occurring

say) performed at time *tk* is independent of any other measurement (

at later time *tj*. Due to the HVT, we must have ( dropping *a*ˆ*k*, *a*ˆ)

between *a*ˆ*<sup>i</sup>* and the positive *z*-axis, *θij* (*i* = 1, 1� and *j* = 2, 2�

<sup>2</sup> <sup>≤</sup> *<sup>E</sup>*(*α*1, *<sup>α</sup>*2, *<sup>λ</sup>*) + *<sup>E</sup>*(*α*1, *<sup>α</sup>*�

measurements of spin-*s* observables on a spin-*s* initial state:

2


Using condition (6) and the result (Shimony, n.d.) (Jarrett, 1984)

*y* − *x*�

*<sup>E</sup>*(*αi*, *<sup>α</sup>j*, *<sup>λ</sup>*) = <sup>∑</sup>*<sup>α</sup>i*,*α<sup>j</sup>*

�*αiαj*� =

<sup>1</sup>} and {*a*ˆ2, *a*ˆ�

*y*� ≤ 2*s*


*<sup>α</sup>iα<sup>j</sup> <sup>p</sup>*(*αi*, *<sup>α</sup>j*|*λ*) = <sup>∑</sup>*<sup>α</sup><sup>i</sup>*

by equation (4). Now let us consider the case of two successive measurements, with options

2, *x*, *y*, *x*�

2, *λ*) + *E*(*α*�

Multiplying by *ρ*(*λ*)*dλ* and integrating over Λ, we get the CHSH-type inequality (Clauser et al., 1969) (involving the hidden variable *λ*) corresponding to performing two successive

<sup>2</sup>� + �*α*�

<sup>2</sup> respectively for measuring spin components. In each run of the experiment,

*p*(*αi*, *a*ˆ*k*&*αj*, *a*ˆ|*λ*) = *p*(*αi*, *a*ˆ*k*|*λ*)*p*(*αj*, *a*ˆ*l*|*λ*). (4)

*S* · *a*ˆ*<sup>k</sup>*

*S*.*a*ˆ*<sup>j</sup>*

) to be the angle

2. (7)

2. (8)

) is the angle between *a*ˆ*<sup>j</sup>* and *a*ˆ*i*.

*S* · *a*ˆ*<sup>l</sup>* say) made at some

a stochastic hidden variable *λ* so that

in the previous measurements.

observable

where

*a*ˆ1, *a*ˆ�

we obtain

<sup>1</sup> and *a*ˆ2, *a*ˆ�

−2*s*

− 2*s*

a random choice between {*a*ˆ1, *a*ˆ�

<sup>2</sup> <sup>≤</sup> *xy* <sup>+</sup> *xy*� <sup>+</sup> *<sup>x</sup>*�

2004).

$$\{x, y, z, x', y', z' \in \{-s, -s + 1, \dots, s - 1, s\}\}$$

and1

$$E(\mathfrak{a}\_{\dot{\nu}}, \mathfrak{a}\_{\dot{\nu}}, \mathfrak{a}\_{k'} \lambda) = E(\mathfrak{a}\_{\dot{\nu}} \lambda) E(\mathfrak{a}\_{\dot{\nu}}, \lambda) E(\mathfrak{a}\_{k'} \lambda)\_{\prime}$$

we can prove Mermin-Klyshko Inequality (MKI) (Mermin, 1990), (Belinskii & Klyshko, 1993) for three successive measurements,

$$|\langle MKI \rangle| = \frac{1}{2} |\langle a\_1 a\_2 a\_3' \rangle + \langle a\_1 a\_2' a\_3 \rangle + \langle a\_1' a\_2 a\_3 \rangle - \langle a\_1' a\_2' a\_3' \rangle| \le s^3. \tag{9}$$

Let |�*MKI*� �| ≤ *<sup>s</sup>*3, where |�*MKI*� �| is obtained from equation (9) by interchanging primes with non-primes in MKI. It is easily shown that

$$|\langle \mathcal{S}I \rangle| = |\langle MKI \rangle + \langle MKI' \rangle| \le |\langle MKI \rangle| + |\langle MKI' \rangle| \le 2s^3. \tag{10}$$

This is the Svetlichny inequality (SI) (Svetlichny, 1987),(Seevinck & Svetlichny, 2002),(Collins, Gisin, Popescu, Roberts & Scarani, 2002).

For *n* successive measurements on spin *s* system, we define the MK polynomials recursively as follows:

$$M\_1 = \mathfrak{a}\_{1\prime} \ M\_1^{\prime} = \mathfrak{a}\_{1\prime}^{\prime} \tag{11}$$

$$M\_{\rm ll} = \frac{1}{2} M\_{\rm n-1} (\boldsymbol{a}\_{\rm nl} + \boldsymbol{a}\_{\rm nl}') + \frac{1}{2} M\_{\rm n-1}' (\boldsymbol{a}\_{\rm nl} - \boldsymbol{a}\_{\rm n}'),\tag{12}$$

where *M*� *<sup>n</sup>* are obtained from *Mn* by interchanging all primed and non-primed *α*'s. The recursive relation (12) gives, for all 1 ≤ *k* ≤ *n* − 1 (Collins, Gisin, Popescu, Roberts & Scarani, 2002) ,(Cabello, 2002a):

$$M\_{\hbar} = \frac{1}{2} M\_{\hbar - k} (M\_{\hbar} + M\_{k}') + \frac{1}{2} M\_{n-k}' (M\_{\hbar} - M\_{k}'). \tag{13}$$

In particular, we have

$$M\_2 = BI = \frac{1}{2}(\alpha\_1 \alpha\_2 + \alpha\_1' \alpha\_2 + \alpha\_1 \alpha\_2' - \alpha\_1' \alpha\_2'),\tag{14}$$

$$M\_3 = MKI = \frac{1}{2} (a\_1 a\_2 a\_3' + a\_1 a\_2' a\_3 + a\_1' a\_2 a\_3 - a\_1' a\_2' a\_3'). \tag{15}$$

We now show that in HVT,

$$|\langle M\_{\hbar} \rangle| \le s^{\hbar}.\tag{16}$$

<sup>1</sup> This is obtained by using equation (4) and the similar argument as has been used in deriving equation (6).

Spin Measurements 9

Quantum Correlations in Successive Spin Measurements 205

performs measurement along *a*ˆ2, one of the values *α*<sup>2</sup> ∈ {−*s*, −*s* + 1, . . . ,*s* − 1,*s*} will arise.

*<sup>S</sup>* · *<sup>a</sup>*ˆ0, *<sup>α</sup>*0|�

the probabilities factorize like those of a Markov chain (Beck & Graudenz, 1992). Every factor in (20) corresponds to the transition amplitude between two successive measurements. By

It is to be noted that although *A* can have positive and negative values, *B* will always be positive. Moreover, for all *<sup>θ</sup>* <sup>∈</sup> [0, 2*π*], if *<sup>A</sup>* <sup>≥</sup> 0, *<sup>A</sup>*cos2*θ*<sup>1</sup> <sup>+</sup> *<sup>B</sup>* is always positive and if *<sup>A</sup>* <sup>&</sup>lt; 0, *<sup>A</sup>*cos2*θ*<sup>1</sup> <sup>+</sup> *<sup>B</sup>* <sup>≥</sup> *<sup>B</sup>* <sup>+</sup> *<sup>A</sup>* <sup>=</sup> <sup>2</sup>*<sup>χ</sup>* <sup>≥</sup> 0. We now have the following expression for the quantity *BI*,

cos*θ*<sup>12</sup> + cos*θ*�

12 + cos*θ*��

expressions for *A* and *B* in terms of *χ* and *s*. Note that the second term in equation (22) (*i.e.*, the

corresponding to the CHSH inequality (Clauser et al., 1969). And hence, its maximum value

have to take care about maximization of the first term in equation (22) and that might require these four vectors to be on different planes. In order to resolve this issue, we now consider

<sup>1</sup>, *φ*�

<sup>4</sup> ) is similar to the expression for �*α*1*α*2� + �*α*1*α*�

<sup>12</sup> is the angle between *a*ˆ�

<sup>2</sup> ∈ [0, *π*] and *φ*1, *φ*�

<sup>2</sup>)] + <sup>1</sup> 4

<sup>1</sup> − *φ*2) − sin *θ*�

(*A*cos2*θ*<sup>1</sup> + *B*)[cos *θ*1(cos *θ*<sup>2</sup> + cos *θ*�

<sup>2</sup> cos(*φ*<sup>1</sup> − *φ*�

<sup>12</sup>)+(*<sup>A</sup>* cos<sup>2</sup> *<sup>θ</sup>*�

+ cos2*θ*� 1 cos*θ*��

<sup>12</sup> − cos*θ*���

<sup>1</sup> and *a*ˆ��

<sup>1</sup>, *a*ˆ2, *a*ˆ�

<sup>1</sup>), (*θ*2, *φ*2), (*θ*�

<sup>1</sup>, *φ*2, *φ*�

(*A* cos<sup>2</sup> *θ*�

<sup>1</sup> sin *θ*�

12

*<sup>S</sup>* · *<sup>a</sup>*ˆ1, *<sup>α</sup>*1�|2|��

*S* · *a*ˆ*<sup>i</sup>* are complete observables, all of whose eigenvalues are non degenerate,

*<sup>S</sup>* · *<sup>a</sup>*ˆ1, *<sup>α</sup>*1|�

<sup>2</sup> cos *<sup>θ</sup>*12[*<sup>A</sup>* cos<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> <sup>+</sup> *<sup>B</sup>*], (21)

+*s* ∑ *α*0=−*s*

<sup>1</sup> + *B*)(cos *θ*��

<sup>12</sup> is the angle between *a*ˆ1 and *a*ˆ�

<sup>2</sup>, *φ*�

<sup>1</sup> + *B*)[cos *θ*�

<sup>1</sup> − *φ*�

<sup>2</sup> cos(*φ*�

<sup>12</sup> − cos *θ*���

12

<sup>12</sup> , (22)

<sup>2</sup> . We have used, in Eq (22), the

<sup>2</sup>� + �*α*�

<sup>2</sup> on the same plane. But we also

<sup>2</sup>) of the vectors *a*ˆ1, *a*ˆ�

<sup>1</sup>(cos *θ*<sup>2</sup> − cos *θ*�

2)

<sup>2</sup>)]. (23)

<sup>2</sup> ∈ [0, 2*π*]. Then *BI* has the form

<sup>2</sup>) + sin *θ*<sup>1</sup> sin *θ*<sup>2</sup> cos(*φ*<sup>1</sup> − *φ*2)

<sup>12</sup> − cos*θ*���

<sup>12</sup>)}

<sup>1</sup> is the angle between

<sup>2</sup>, *θ*��

1*α*2�−�*α*�

<sup>12</sup> is the

1*α*� 2�

<sup>1</sup>, *a*ˆ2, *a*ˆ� 2

*α*2 <sup>0</sup> *pα*<sup>0</sup> .

*<sup>S</sup>* · *<sup>a</sup>*ˆ2, *<sup>α</sup>*2�|2.

(20)

So

Note that, since �

where

equation (A.12), we get

appeared in equation (8):

<sup>=</sup> <sup>3</sup>*<sup>χ</sup>* <sup>−</sup> *<sup>s</sup>*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) 4

> *s*(*s* + 1) − *χ* 4

> > <sup>1</sup> and *a*ˆ2, *θ*���

the spherical-polar co-ordinates (*θ*1, *φ*1), (*θ*�

+ sin *θ*<sup>1</sup> sin *θ*�

+ sin *θ*�

*BI* <sup>=</sup> <sup>1</sup> 4

+

term with the factor *<sup>s</sup>*(*s*+1)−*<sup>χ</sup>*

respectively, where *θ*1, *θ*�

*BI* <sup>=</sup> <sup>1</sup> 4

*a*ˆ0 and *a*ˆ�

angle between *a*ˆ�

�*α*1*α*2�*QM* = *Tr*(*ρ*<sup>1</sup>

� *S* · *a*ˆ1 � *S* · *a*ˆ2) =

*<sup>p</sup>α*0*α*1*α*2|��

�*α*1*α*2� <sup>=</sup> <sup>1</sup>

{(*<sup>A</sup>* cos<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> <sup>+</sup> *<sup>B</sup>*)(cos *<sup>θ</sup>*<sup>12</sup> <sup>+</sup> cos *<sup>θ</sup>*�

cos*θ*<sup>12</sup> + cos*θ*�

where (according to Appendix A) *θ*<sup>1</sup> is the angle between *a*ˆ0 and *a*ˆ1, *θ*�

 cos2*θ*<sup>1</sup> 

<sup>1</sup>, *θ*<sup>12</sup> is the angle between *a*ˆ1 and *a*ˆ2, *θ*�

will occur when we choose all the four vectors *a*ˆ1, *a*ˆ�

<sup>1</sup>, *θ*2, *θ*�

<sup>1</sup> sin *θ*<sup>2</sup> cos(*φ*�

*A* = 3*χ* − *s*(*s* + 1), *B* = *s*(*s* + 1) − *χ*, *χ* =

<sup>∑</sup>*<sup>α</sup>*0*α*1*α*<sup>2</sup>

First note that (16) is true for *n* = 2, 3 (equations (8), (9)). Suppose it is true for *n* = *k* i.e. *Max*|�*Mk*�| <sup>=</sup> *<sup>s</sup>k*. Now

$$|\langle M\_{k+1}\rangle| = \frac{1}{2}|\langle M\_k\alpha\_{k+1}\rangle + \langle M\_k\alpha'\_{k+1}\rangle + \langle M'\_k\alpha\_{k+1}\rangle - \langle M'\_k\alpha'\_{k+1}\rangle|.$$

Since HVT applies here we can use (4) to get

$$|\langle M\_{k+1}\rangle| = \frac{1}{2} |\langle M\_k\rangle (\langle a\_{k+1}\rangle + \langle a'\_{k+1}\rangle) + \langle M\_k'\rangle (\langle a\_{k+1}\rangle - \langle a'\_{k+1}\rangle)|.$$

This implies, by induction hypothesis (and using the fact that max|�*M*2�| <sup>=</sup> *<sup>s</sup>*2), that

$$\max |\langle M\_{k+1} \rangle| = s \max |\langle M\_k \rangle| = s^{k+1}.$$

This result is derived for *n* spin-*s* particles by Cabello (Cabello, 2002a).

We now define a quantity, denoted by *ηn*, which will be required later on. *ηn*, is the ratio between maximum |�*Mn*�| given by quantum correlation between *n* successive measurement's outputs and the maximal classical one,

$$\eta\_n = \frac{\max |\langle M\_{\rm n} \rangle\_{QM}|}{s^n}. \tag{17}$$

#### **4. Mixed input state for arbitrary spin**

#### **4.1 Two successive measurements (BI)**

We first deal with the case when input state is a mixed state whose eigenstates coincide with those of � *S* · *a*ˆ0 for some *a*ˆ0 whose eigenvalues we denote by *α*<sup>0</sup> ∈ {−*s*, ···*s*}. For spin 1/2 this is the most general mixed state because given any density operator *ρ*<sup>0</sup> for spin 1/2 (corresponding to some point within the Bloch sphere), we can find an *a*ˆ0 such that the eigenstates of � *S*·, *a*ˆ0 and *ρ*<sup>0</sup> coincide. However, for *s* > 1/2, our choice forms a restricted class of mixed states. We note that these are the only states accessible via SG experiments. Thus we have

$$\rho\_0 = \sum\_{a\_0} p\_{a\_0} |\vec{S} \cdot \mathfrak{a}\_0, a\_0\rangle \langle \vec{S} \cdot \mathfrak{a}\_0, a\_0|; \qquad \left(\sum\_{a\_0} p\_{a\_0} = 1\right) \tag{18}$$

After the first measurement along *a*ˆ1, the resulting state of the system is

$$
\rho\_1 = \sum\_{\mathfrak{a}\_1} M\_{\mathfrak{a}\_1}^\dagger \rho\_0 M\_{\mathfrak{a}\_1 \mathfrak{a}\_1} \tag{19}
$$

where

$$M\_{\mathfrak{a}\_1}^\dagger = M\_{\mathfrak{a}\_1} = |\vec{S} \cdot \mathfrak{a}\_1 \,\, \alpha\_1 \rangle \langle \vec{S} \cdot \mathfrak{a}\_1 \,\, \alpha\_1 | \,.$$

Now �*α*1*α*2�*QM* is the expectation value (according to QM) that given the initial state *ρ*<sup>0</sup> (given in equation (18)), the 1st measurement along *a*ˆ1 will give rise to any value *α*<sup>1</sup> ∈ {−*s*, −*s* + 1, . . . ,*s* − 1,*s*}, and then, on the after-measurement state *ρ*<sup>1</sup> (given in equation (19)), if one performs measurement along *a*ˆ2, one of the values *α*<sup>2</sup> ∈ {−*s*, −*s* + 1, . . . ,*s* − 1,*s*} will arise. So

$$\begin{split} \langle \mathfrak{a}\_{1}\mathfrak{a}\_{2} \rangle\_{QM} &= \operatorname{Tr}(\rho\_{1}\vec{S}\cdot\mathfrak{a}\_{1}\vec{S}\cdot\mathfrak{A}\_{2}) = \\ &\sum\_{\mathfrak{a}\_{0}\mathfrak{a}\_{1}\mathfrak{a}\_{2}} p\_{\mathfrak{a}\_{0}\mathfrak{a}\_{1}\mathfrak{a}\_{2}} |\langle \vec{S}\cdot\hat{\mathfrak{a}}\_{0},\mathfrak{a}\_{0}|\vec{S}\cdot\hat{\mathfrak{a}}\_{1},\mathfrak{a}\_{1}\rangle|^{2} |\langle \vec{S}\cdot\hat{\mathfrak{a}}\_{1},\mathfrak{a}\_{1}|\vec{S}\cdot\hat{\mathfrak{a}}\_{2},\mathfrak{a}\_{2}\rangle|^{2} . \end{split} \tag{20}$$

Note that, since � *S* · *a*ˆ*<sup>i</sup>* are complete observables, all of whose eigenvalues are non degenerate, the probabilities factorize like those of a Markov chain (Beck & Graudenz, 1992). Every factor in (20) corresponds to the transition amplitude between two successive measurements. By equation (A.12), we get

$$
\langle a\_1 a\_2 \rangle = \frac{1}{2} \cos \theta\_{12} [A \cos^2 \theta\_1 + B] \,\tag{21}
$$

where

8 Will-be-set-by-IN-TECH

First note that (16) is true for *n* = 2, 3 (equations (8), (9)). Suppose it is true for *n* = *k* i.e.

*<sup>k</sup>*+1� + �*M*�

*<sup>k</sup>*+1�) + �*M*�

*kαk*+1�−�*M*�

*k*�(�*αk*+1�−�*α*�

*<sup>k</sup>*<sup>+</sup>1.

*kα*� *<sup>k</sup>*+1�|.

*<sup>s</sup><sup>n</sup>* . (17)

*<sup>k</sup>*+1�)|.



This result is derived for *n* spin-*s* particles by Cabello (Cabello, 2002a).

This implies, by induction hypothesis (and using the fact that max|�*M*2�| <sup>=</sup> *<sup>s</sup>*2), that

max |�*Mk*<sup>+</sup>1�| = *s* max |�*Mk*�| = *s*

We now define a quantity, denoted by *ηn*, which will be required later on. *ηn*, is the ratio between maximum |�*Mn*�| given by quantum correlation between *n* successive

*<sup>η</sup><sup>n</sup>* <sup>=</sup> *max*|�*Mn*�*QM*<sup>|</sup>

We first deal with the case when input state is a mixed state whose eigenstates coincide

spin 1/2 this is the most general mixed state because given any density operator *ρ*<sup>0</sup> for spin 1/2 (corresponding to some point within the Bloch sphere), we can find an *a*ˆ0 such that the

of mixed states. We note that these are the only states accessible via SG experiments. Thus we

*S* · *a*ˆ0, *α*0|;

*M*†

Now �*α*1*α*2�*QM* is the expectation value (according to QM) that given the initial state *ρ*<sup>0</sup> (given in equation (18)), the 1st measurement along *a*ˆ1 will give rise to any value *α*<sup>1</sup> ∈ {−*s*, −*s* + 1, . . . ,*s* − 1,*s*}, and then, on the after-measurement state *ρ*<sup>1</sup> (given in equation (19)), if one

*<sup>S</sup>* · *<sup>a</sup>*ˆ1, *<sup>α</sup>*1���

*S* · *a*ˆ0 for some *a*ˆ0 whose eigenvalues we denote by *α*<sup>0</sup> ∈ {−*s*, ···*s*}. For

*S*·, *a*ˆ0 and *ρ*<sup>0</sup> coincide. However, for *s* > 1/2, our choice forms a restricted class

 ∑*α*0

*S* · *a*ˆ1, *α*1|.

*pα*<sup>0</sup> = 1

*<sup>α</sup>*<sup>1</sup> *ρ*0*Mα*<sup>1</sup> , (19)

(18)

*Max*|�*Mk*�| <sup>=</sup> *<sup>s</sup>k*. Now


Since HVT applies here we can use (4) to get


**4. Mixed input state for arbitrary spin**

*<sup>ρ</sup>*<sup>0</sup> <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*<sup>0</sup>

*<sup>p</sup>α*<sup>0</sup> <sup>|</sup>�

*M*†

*<sup>S</sup>* · *<sup>a</sup>*ˆ0, *<sup>α</sup>*0���

*<sup>ρ</sup>*<sup>1</sup> <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*<sup>1</sup>

*<sup>α</sup>*<sup>1</sup> <sup>=</sup> *<sup>M</sup>α*<sup>1</sup> <sup>=</sup> <sup>|</sup>�

After the first measurement along *a*ˆ1, the resulting state of the system is

**4.1 Two successive measurements (BI)**

with those of �

eigenstates of �

have

where

2

2

measurement's outputs and the maximal classical one,

$$A = 3\chi - s(s+1), \quad B = s(s+1) - \chi\_{\prime} \quad \chi = \sum\_{\alpha\_0=-s}^{+s} a\_0^2 p\_{\alpha\_0}.$$

It is to be noted that although *A* can have positive and negative values, *B* will always be positive. Moreover, for all *<sup>θ</sup>* <sup>∈</sup> [0, 2*π*], if *<sup>A</sup>* <sup>≥</sup> 0, *<sup>A</sup>*cos2*θ*<sup>1</sup> <sup>+</sup> *<sup>B</sup>* is always positive and if *<sup>A</sup>* <sup>&</sup>lt; 0, *<sup>A</sup>*cos2*θ*<sup>1</sup> <sup>+</sup> *<sup>B</sup>* <sup>≥</sup> *<sup>B</sup>* <sup>+</sup> *<sup>A</sup>* <sup>=</sup> <sup>2</sup>*<sup>χ</sup>* <sup>≥</sup> 0. We now have the following expression for the quantity *BI*, appeared in equation (8):

$$\begin{split} BI &= \frac{1}{4} \{ (A\cos^2\theta\_1 + B)(\cos\theta\_{12} + \cos\theta\_{12}') + (A\cos^2\theta\_1' + B)(\cos\theta\_{12}' - \cos\theta\_{12}^{\prime\prime\prime}) \} \\ &= \frac{3\chi - s(s+1)}{4} \left\{ \cos^2\theta\_1 \left( \cos\theta\_{12} + \cos\theta\_{12}' \right) + \cos^2\theta\_1' \left( \cos\theta\_{12}^{\prime\prime} - \cos\theta\_{12}^{\prime\prime\prime} \right) \right\} \\ &+ \frac{s(s+1) - \chi}{4} \left\{ \left( \cos\theta\_{12} + \cos\theta\_{12}' \right) + \left( \cos\theta\_{12}^{\prime\prime} - \cos\theta\_{12}^{\prime\prime\prime} \right) \right\}, \end{split} \tag{22}$$

where (according to Appendix A) *θ*<sup>1</sup> is the angle between *a*ˆ0 and *a*ˆ1, *θ*� <sup>1</sup> is the angle between *a*ˆ0 and *a*ˆ� <sup>1</sup>, *θ*<sup>12</sup> is the angle between *a*ˆ1 and *a*ˆ2, *θ*� <sup>12</sup> is the angle between *a*ˆ1 and *a*ˆ� <sup>2</sup>, *θ*�� <sup>12</sup> is the angle between *a*ˆ� <sup>1</sup> and *a*ˆ2, *θ*��� <sup>12</sup> is the angle between *a*ˆ� <sup>1</sup> and *a*ˆ�� <sup>2</sup> . We have used, in Eq (22), the expressions for *A* and *B* in terms of *χ* and *s*. Note that the second term in equation (22) (*i.e.*, the term with the factor *<sup>s</sup>*(*s*+1)−*<sup>χ</sup>* <sup>4</sup> ) is similar to the expression for �*α*1*α*2� + �*α*1*α*� <sup>2</sup>� + �*α*� 1*α*2�−�*α*� 1*α*� 2� corresponding to the CHSH inequality (Clauser et al., 1969). And hence, its maximum value will occur when we choose all the four vectors *a*ˆ1, *a*ˆ� <sup>1</sup>, *a*ˆ2, *a*ˆ� <sup>2</sup> on the same plane. But we also have to take care about maximization of the first term in equation (22) and that might require these four vectors to be on different planes. In order to resolve this issue, we now consider the spherical-polar co-ordinates (*θ*1, *φ*1), (*θ*� <sup>1</sup>, *φ*� <sup>1</sup>), (*θ*2, *φ*2), (*θ*� <sup>2</sup>, *φ*� <sup>2</sup>) of the vectors *a*ˆ1, *a*ˆ� <sup>1</sup>, *a*ˆ2, *a*ˆ� 2 respectively, where *θ*1, *θ*� <sup>1</sup>, *θ*2, *θ*� <sup>2</sup> ∈ [0, *π*] and *φ*1, *φ*� <sup>1</sup>, *φ*2, *φ*� <sup>2</sup> ∈ [0, 2*π*]. Then *BI* has the form

$$BI = \frac{1}{4}(A\cos^2\theta\_1 + B)[\cos\theta\_1(\cos\theta\_2 + \cos\theta\_2') + \sin\theta\_1\sin\theta\_2\cos(\phi\_1 - \phi\_2)]$$

$$\begin{split} &+ \sin\theta\_1\sin\theta\_2'\cos(\phi\_1 - \phi\_2')) + \frac{1}{4}(A\cos^2\theta\_1' + B)[\cos\theta\_1'(\cos\theta\_2 - \cos\theta\_2') \\ &+ \sin\theta\_1'\sin\theta\_2\cos(\phi\_1' - \phi\_2) - \sin\theta\_1'\sin\theta\_2'\cos(\phi\_1' - \phi\_2')]. \end{split} \tag{23}$$

Spin Measurements 11

Quantum Correlations in Successive Spin Measurements 207

Thus we see that, when *s* = 1, only those input states *ρ*<sup>0</sup> ( given in equation (18)) will break BI for each of which *pα*0=<sup>0</sup> ∈ [0, 0.23) ∪ (0.67, 1]. Next we consider the situations where *s* > 1.

{*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>ξ</sup>s*2}tan3*θ*<sup>1</sup> <sup>+</sup> {7*ξs*<sup>2</sup> <sup>−</sup> <sup>3</sup>*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>)}tan2*θ*1<sup>+</sup>

*<sup>S</sup>* · *<sup>a</sup>*ˆ0,*s*<sup>|</sup> <sup>+</sup> *<sup>p</sup>*−*s*|�

with *ps* + *<sup>p</sup>*−*<sup>s</sup>* = 1 and *ps*−1, *<sup>p</sup>*−*s*+1, *ps*−2, *<sup>p</sup>*−*s*+2,... = 0. Thus we see here that *<sup>ξ</sup>* = 1. Also

As here *s* > 1, therefore the last equation will have only one positive root and the other two

is violated. If *ρ*<sup>0</sup> has contribution from neither of the states corresponding to *α*<sup>0</sup> = ±*s* (*i.e.*,

+ (*ps*−<sup>2</sup> + *p*−*s*+2)

But the quantity <sup>√</sup>2(<sup>1</sup> <sup>−</sup> 1/*s*)<sup>2</sup> is less than 1 for all *<sup>s</sup>* <sup>=</sup> 1/2, 1, 3/2, . . . , 6. Therefore, for *<sup>s</sup>* <sup>&</sup>gt; 1, if the initial state *ρ*<sup>0</sup> has contribution from neither of the states corresponding to *α*<sup>0</sup> = ±*s*, BI will be satisfied for all *s* ≤ 6. Thus we see that whenever *s* ∈ {3/2, 2, 5/2, . . . , 6}, in order that *ρ*<sup>0</sup> violates BI, the associated quantity *ξ* must have values near 1. In table 1, we have given the ranges of values of *ξ* (obtained numerically) for which BI is violated, starting from *s* = 1/2.

<sup>×</sup> <sup>1</sup> <sup>×</sup> [(*<sup>s</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>s</sup>ξ*)+(3*s<sup>ξ</sup>* <sup>−</sup> *<sup>s</sup>* <sup>−</sup> <sup>1</sup>) <sup>×</sup> <sup>1</sup>] = <sup>√</sup>

+ (*ps*−<sup>2</sup> + *p*−*s*+2)

<sup>2</sup> <sup>−</sup> *<sup>s</sup>*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>)}cos2*θ*<sup>1</sup> <sup>+</sup> {*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>ξ</sup><sup>s</sup>*

*<sup>S</sup>* · *<sup>a</sup>*ˆ0, <sup>−</sup>*s*���

*<sup>η</sup>*<sup>2</sup> = (1/2*s*)|sin*θ*<sup>1</sup> <sup>+</sup> cos*θ*1|{(2*<sup>s</sup>* <sup>−</sup> <sup>1</sup>)cos2*θ*<sup>1</sup> <sup>+</sup> <sup>1</sup>}, (29)

tan3*θ*<sup>1</sup> + (4*<sup>s</sup>* <sup>−</sup> <sup>3</sup>)tan2*θ*<sup>1</sup> + (6*<sup>s</sup>* <sup>−</sup> <sup>2</sup>)tan*θ*<sup>1</sup> <sup>−</sup> <sup>2</sup>*<sup>s</sup>* <sup>=</sup> 0. (30)

 <sup>1</sup> <sup>−</sup> <sup>2</sup> *s* <sup>2</sup>

(*<sup>s</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>s</sup>ξ*)+(3*s<sup>ξ</sup>* <sup>−</sup> *<sup>s</sup>* <sup>−</sup> <sup>1</sup>)cos2*θ*<sup>1</sup>

<sup>2</sup>*<sup>ξ</sup>* <sup>≤</sup> <sup>√</sup> 2 <sup>1</sup> <sup>−</sup> <sup>1</sup> *s* <sup>2</sup> .

<sup>1</sup> (*s*)) > 1 for all *s* > 1. Hence, in this case, BI

+ ... ≤

 <sup>1</sup> <sup>−</sup> <sup>1</sup> *s* <sup>2</sup> .

{8*ξs*<sup>2</sup> <sup>−</sup> <sup>2</sup>*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>)}tan*θ*<sup>1</sup> <sup>−</sup> <sup>2</sup>*ξs*<sup>2</sup> <sup>=</sup> 0. (27)

 <sup>1</sup> <sup>−</sup> <sup>2</sup> *s* <sup>2</sup>

*<sup>α</sup>*0=−*<sup>s</sup> <sup>p</sup>α*<sup>0</sup> = 1. Therefore, we must

2} 

*S* · *a*ˆ0, −*s*|, (28)

<sup>1</sup> (*s*) ∈ (0, *π*/4) for

+ ...,

, (26)

 <sup>1</sup> <sup>−</sup> <sup>1</sup> *s* <sup>2</sup>

have 0 ≤ *ξ* ≤ 1. In this case, equations (24) and (25) respectively take the forms

 {3*ξs*

*<sup>S</sup>* · *<sup>a</sup>*ˆ0,*s*���

equations (26) and (27) have respectively been turned into the forms

roots will be complex. The positive root will correspond to an angle *θmax*

<sup>2</sup> (*s*) <sup>≡</sup> *<sup>η</sup>*2(*θmax*

Note that, by definition (true for all *s*),

*<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup>

which it can be shown that *ηmax*

*<sup>ξ</sup>* = (*ps*−<sup>1</sup> + *<sup>p</sup>*−*s*+1)

From equation (24) it follows that

*<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>√</sup>2*<sup>s</sup>*  <sup>1</sup> <sup>−</sup> <sup>1</sup> *s* <sup>2</sup>

The case when *s* → ∞ has also been considered in table 1.


*ps* = *p*−*<sup>s</sup>* = 0), we have

< 1 <sup>√</sup>2*<sup>s</sup>*

*<sup>ξ</sup>* = (*ps* + *<sup>p</sup>*−*s*) + (*ps*−<sup>1</sup> + *<sup>p</sup>*−*s*+1)

<sup>2</sup>*s*<sup>2</sup> <sup>|</sup>sin*θ*<sup>1</sup> <sup>+</sup> cos*θ*1<sup>|</sup>

Let us first consider the input states of the form

*<sup>ρ</sup>*<sup>0</sup> <sup>=</sup> *ps*|�

where 0 <sup>≤</sup> *ps*, *<sup>p</sup>*−*s*, *ps*−1, *<sup>p</sup>*−*s*+1, *ps*−2, *<sup>p</sup>*−*s*+2,... <sup>≤</sup> 1 and <sup>∑</sup>*<sup>s</sup>*

Here also the maximum value of |*BI*| will occur when all the vectors *a*ˆ1, *a*ˆ� <sup>1</sup>, *a*ˆ2, *a*ˆ� <sup>2</sup> lie on the same plane. This is obtained by:

$$\frac{\partial BI}{\partial \phi\_1} = \frac{\partial BI}{\partial \phi\_2} = \frac{\partial BI}{\partial \phi\_1'} = \frac{\partialBI}{\partial \phi\_2'} = 0 \Rightarrow \phi\_1 = \phi\_1' = \phi\_2 = \phi\_2'.$$

In that case, the maximum value of |*BI*| will occur when *θ*� <sup>1</sup> = *π* − *θ*1, *θ*<sup>2</sup> = *π*/2, *θ*� <sup>2</sup> = 0 and (correspondingly) the quantity

$$\eta\_2 = \frac{|BI|}{s^2} = \left(\frac{1}{2s^2}\right) |(\sin\theta\_1 + \cos\theta\_1)|(A\cos^2\theta\_1 + B) \tag{24}$$

is maximized over all possible values of *<sup>θ</sup>*<sup>1</sup> (*<sup>A</sup>* cos<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> <sup>+</sup> *<sup>B</sup>* <sup>≥</sup> 0). If *<sup>η</sup>*<sup>2</sup> <sup>&</sup>gt; 1, the correlations for two successive measurements violate the CHSH-type inequality (8), and hence a contradiction with the above-mentioned HVT. In fact *∂η*<sup>2</sup> *∂θ*1 = 0 implies that

$$B\tan^3\theta\_1 + (2A - B)\tan^2\theta\_1 + (3A + B)\tan\theta\_1 - (A + B) = 0.\tag{25}$$

Real roots (for tan*θ*1) of this equation give values of *θ*<sup>1</sup> for which *η*<sup>2</sup> is maximum. The maximum value of *η*<sup>2</sup> is evaluated at these *θ*1's.

We find that for *s* = <sup>1</sup> <sup>2</sup> , *χ* = 1/4 for all *ρ*0, and so *A* = 0, *B* = 1/4. So, from equation (24), we have *<sup>η</sup>*<sup>2</sup> <sup>=</sup> sin*θ*<sup>1</sup> <sup>+</sup> cos*θ*1. Therefore equation (25) becomes tan3*θ*<sup>1</sup> <sup>−</sup> tan2*θ*<sup>1</sup> <sup>+</sup> tan*θ*<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>=</sup> 0, whose only one real solution is tan*θ*<sup>1</sup> = 1. So *θ*<sup>1</sup> = *π*/4 or 5*π*/4. *θ*<sup>1</sup> = *π*/4 gives the maximum possible value *<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>√</sup><sup>2</sup> <sup>&</sup>gt; 1. Thus all possible spin-1/2 states break BI for (proper choices of) two successive measurements. This can be compared with the measurement correlations corresponding to measurement of spin observables on space-like separated two particles scenario where only the entangled pure states break BI while not all entangled mixed states break it (Werner, 1989).

From now on, we will use the range of values of the quantity *<sup>ξ</sup>* <sup>≡</sup> *<sup>χ</sup>*/*s*<sup>2</sup> to identify the parametric region of the initial density matrix *ρ*<sup>0</sup> where the inequality (8) will be violated. Thus we see that for all spin-1/2 input states *ρ*0, *ξ* = 1.

For a spin-1 system, we first consider all input states *ρ*<sup>0</sup> none of which have a contribution of *Sz* = 0 eigenstate. In this case *χ* = 1, *A* = *B* = 1. So *η*<sup>2</sup> = (1/2)(sin*θ*<sup>1</sup> + cos*θ*1)(cos2*θ*<sup>1</sup> + 1) and equation (25) takes the form tan3*θ*<sup>1</sup> <sup>+</sup> tan2*θ*<sup>1</sup> <sup>+</sup> 4tan3*θ*<sup>1</sup> <sup>−</sup> <sup>2</sup> <sup>=</sup> 0. The only real root of this equation is tan*θ*<sup>1</sup> ≈ 0.433. Thus the maximum possible value of *η*<sup>2</sup> is (using equation (24)) 1.2112 ( approximately). Thus we see that all input spin-1 states *ρ*0, none of which has a component along |*Sz* = 0�, break BI (equation (8)) for proper choice of the observables.

Next, for *<sup>s</sup>* <sup>=</sup> 1, we consider the state *<sup>ρ</sup>*0, for which *<sup>p</sup>α*0=<sup>0</sup> <sup>=</sup> 1, *i.e.*, *<sup>ρ</sup>*<sup>0</sup> <sup>=</sup> <sup>|</sup>� *<sup>S</sup>* · *<sup>a</sup>*ˆ0, 0��� *S* · *a*ˆ0, 0|. In this case, *χ* = 0, and so, *A* = −2, *B* = 2, *ξ* = 0. Then equation (25) takes the form 2tan3*θ*<sup>1</sup> <sup>−</sup> 6tan2*θ*<sup>1</sup> <sup>−</sup> 4tan*θ*<sup>1</sup> <sup>=</sup> 0. It has three real solutions, which corresponds to *<sup>θ</sup>*<sup>1</sup> <sup>=</sup> <sup>0</sup> (or *π*), 74.3165*<sup>o</sup>* (approx.), 150.6836*<sup>o</sup>* (approx.). The maximum possible value of *η*<sup>2</sup> occurs at *<sup>θ</sup>*<sup>1</sup> <sup>=</sup> 74.3165*o*, and the corresponding value is given by *<sup>η</sup>*<sup>2</sup> <sup>≈</sup> 1.1428. Thus the state *<sup>ρ</sup>*<sup>0</sup> <sup>=</sup> |� *<sup>S</sup>* · *<sup>a</sup>*ˆ0, 0��� *S* · *a*ˆ0, 0| breaks the BI (equation (8)).

For *s* = 1, when 0 < *pα*0=<sup>0</sup> ≡ *p*<sup>0</sup> (say) < 1, we have *χ* = 1 − *p*<sup>0</sup> = *ξ*, *A* = 1 − 3*p*<sup>0</sup> = 3*ξ* − 2 and *<sup>B</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*. We then have *<sup>η</sup>*<sup>2</sup> = (1/2)|sin*θ*<sup>1</sup> <sup>+</sup> cos*θ*1|{(3*<sup>ξ</sup>* <sup>−</sup> <sup>2</sup>)cos2*θ*<sup>1</sup> + (<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*)} (by equation (24)), and equation (25) becomes (<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*)tan3*θ*<sup>1</sup> + (7*<sup>ξ</sup>* <sup>−</sup> <sup>6</sup>)tan2*θ*<sup>1</sup> <sup>+</sup> <sup>4</sup>(2*<sup>ξ</sup>* <sup>−</sup> <sup>1</sup>)tan*θ*<sup>1</sup> <sup>−</sup> 2*ξ* = 0. In this case, one can show numerically that the BI will break (*i.e.*, *η*<sup>2</sup> > 1) if and only if either 0 < *ξ* < 0.33 or 0.77 < *ξ* < 1 ( equivalently, either 0.67 < *p*<sup>0</sup> < 1 or 0 < *p*<sup>0</sup> < 0.23). 10 Will-be-set-by-IN-TECH

is maximized over all possible values of *<sup>θ</sup>*<sup>1</sup> (*<sup>A</sup>* cos<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> <sup>+</sup> *<sup>B</sup>* <sup>≥</sup> 0). If *<sup>η</sup>*<sup>2</sup> <sup>&</sup>gt; 1, the correlations for two successive measurements violate the CHSH-type inequality (8), and hence a contradiction

Real roots (for tan*θ*1) of this equation give values of *θ*<sup>1</sup> for which *η*<sup>2</sup> is maximum. The

have *<sup>η</sup>*<sup>2</sup> <sup>=</sup> sin*θ*<sup>1</sup> <sup>+</sup> cos*θ*1. Therefore equation (25) becomes tan3*θ*<sup>1</sup> <sup>−</sup> tan2*θ*<sup>1</sup> <sup>+</sup> tan*θ*<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>=</sup> 0, whose only one real solution is tan*θ*<sup>1</sup> = 1. So *θ*<sup>1</sup> = *π*/4 or 5*π*/4. *θ*<sup>1</sup> = *π*/4 gives the maximum possible value *<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>√</sup><sup>2</sup> <sup>&</sup>gt; 1. Thus all possible spin-1/2 states break BI for (proper choices of) two successive measurements. This can be compared with the measurement correlations corresponding to measurement of spin observables on space-like separated two particles scenario where only the entangled pure states break BI while not all entangled mixed states

From now on, we will use the range of values of the quantity *<sup>ξ</sup>* <sup>≡</sup> *<sup>χ</sup>*/*s*<sup>2</sup> to identify the parametric region of the initial density matrix *ρ*<sup>0</sup> where the inequality (8) will be violated.

For a spin-1 system, we first consider all input states *ρ*<sup>0</sup> none of which have a contribution of *Sz* = 0 eigenstate. In this case *χ* = 1, *A* = *B* = 1. So *η*<sup>2</sup> = (1/2)(sin*θ*<sup>1</sup> + cos*θ*1)(cos2*θ*<sup>1</sup> + 1) and equation (25) takes the form tan3*θ*<sup>1</sup> <sup>+</sup> tan2*θ*<sup>1</sup> <sup>+</sup> 4tan3*θ*<sup>1</sup> <sup>−</sup> <sup>2</sup> <sup>=</sup> 0. The only real root of this equation is tan*θ*<sup>1</sup> ≈ 0.433. Thus the maximum possible value of *η*<sup>2</sup> is (using equation (24)) 1.2112 ( approximately). Thus we see that all input spin-1 states *ρ*0, none of which has a component along |*Sz* = 0�, break BI (equation (8)) for proper choice of the observables.

In this case, *χ* = 0, and so, *A* = −2, *B* = 2, *ξ* = 0. Then equation (25) takes the form 2tan3*θ*<sup>1</sup> <sup>−</sup> 6tan2*θ*<sup>1</sup> <sup>−</sup> 4tan*θ*<sup>1</sup> <sup>=</sup> 0. It has three real solutions, which corresponds to *<sup>θ</sup>*<sup>1</sup> <sup>=</sup> <sup>0</sup> (or *π*), 74.3165*<sup>o</sup>* (approx.), 150.6836*<sup>o</sup>* (approx.). The maximum possible value of *η*<sup>2</sup> occurs at *<sup>θ</sup>*<sup>1</sup> <sup>=</sup> 74.3165*o*, and the corresponding value is given by *<sup>η</sup>*<sup>2</sup> <sup>≈</sup> 1.1428. Thus the state *<sup>ρ</sup>*<sup>0</sup> <sup>=</sup>

For *s* = 1, when 0 < *pα*0=<sup>0</sup> ≡ *p*<sup>0</sup> (say) < 1, we have *χ* = 1 − *p*<sup>0</sup> = *ξ*, *A* = 1 − 3*p*<sup>0</sup> = 3*ξ* − 2 and *<sup>B</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>p</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*. We then have *<sup>η</sup>*<sup>2</sup> = (1/2)|sin*θ*<sup>1</sup> <sup>+</sup> cos*θ*1|{(3*<sup>ξ</sup>* <sup>−</sup> <sup>2</sup>)cos2*θ*<sup>1</sup> + (<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*)} (by equation (24)), and equation (25) becomes (<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*)tan3*θ*<sup>1</sup> + (7*<sup>ξ</sup>* <sup>−</sup> <sup>6</sup>)tan2*θ*<sup>1</sup> <sup>+</sup> <sup>4</sup>(2*<sup>ξ</sup>* <sup>−</sup> <sup>1</sup>)tan*θ*<sup>1</sup> <sup>−</sup> 2*ξ* = 0. In this case, one can show numerically that the BI will break (*i.e.*, *η*<sup>2</sup> > 1) if and only if either 0 < *ξ* < 0.33 or 0.77 < *ξ* < 1 ( equivalently, either 0.67 < *p*<sup>0</sup> < 1 or 0 < *p*<sup>0</sup> < 0.23).

Next, for *<sup>s</sup>* <sup>=</sup> 1, we consider the state *<sup>ρ</sup>*0, for which *<sup>p</sup>α*0=<sup>0</sup> <sup>=</sup> 1, *i.e.*, *<sup>ρ</sup>*<sup>0</sup> <sup>=</sup> <sup>|</sup>�

= 0 implies that

*<sup>B</sup>* tan<sup>3</sup> *<sup>θ</sup>*<sup>1</sup> + (2*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*)tan2 *<sup>θ</sup>*<sup>1</sup> + (3*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*)tan *<sup>θ</sup>*<sup>1</sup> <sup>−</sup> (*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*) = 0. (25)

<sup>2</sup> , *χ* = 1/4 for all *ρ*0, and so *A* = 0, *B* = 1/4. So, from equation (24), we

*∂θ*1

= 0 ⇒ *φ*<sup>1</sup> = *φ*�

<sup>1</sup> = *φ*<sup>2</sup> = *φ*�

<sup>|</sup>(sin *<sup>θ</sup>*<sup>1</sup> <sup>+</sup> cos *<sup>θ</sup>*1)|(*<sup>A</sup>* cos<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> <sup>+</sup> *<sup>B</sup>*) (24)

2.

<sup>1</sup> = *π* − *θ*1, *θ*<sup>2</sup> = *π*/2, *θ*�

<sup>1</sup>, *a*ˆ2, *a*ˆ�

*<sup>S</sup>* · *<sup>a</sup>*ˆ0, 0���

*S* · *a*ˆ0, 0|.

<sup>2</sup> lie on the

<sup>2</sup> = 0 and

Here also the maximum value of |*BI*| will occur when all the vectors *a*ˆ1, *a*ˆ�

<sup>=</sup> *<sup>∂</sup>BI ∂φ*� 2

<sup>=</sup> *<sup>∂</sup>BI ∂φ*� 1

 1 2*s*<sup>2</sup> 

same plane. This is obtained by:

(correspondingly) the quantity

We find that for *s* = <sup>1</sup>

break it (Werner, 1989).


*<sup>S</sup>* · *<sup>a</sup>*ˆ0, 0���

*∂BI ∂φ*1

*<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>|</sup>*BI*<sup>|</sup>

with the above-mentioned HVT. In fact *∂η*<sup>2</sup>

maximum value of *η*<sup>2</sup> is evaluated at these *θ*1's.

Thus we see that for all spin-1/2 input states *ρ*0, *ξ* = 1.

*S* · *a*ˆ0, 0| breaks the BI (equation (8)).

<sup>=</sup> *<sup>∂</sup>BI ∂φ*2

In that case, the maximum value of |*BI*| will occur when *θ*�

*<sup>s</sup>*<sup>2</sup> <sup>=</sup>

Thus we see that, when *s* = 1, only those input states *ρ*<sup>0</sup> ( given in equation (18)) will break BI for each of which *pα*0=<sup>0</sup> ∈ [0, 0.23) ∪ (0.67, 1]. Next we consider the situations where *s* > 1. Note that, by definition (true for all *s*),

$$\xi = (p\_s + p\_{-s}) + (p\_{s-1} + p\_{-s+1})\left(1 - \frac{1}{s}\right)^2 + (p\_{s-2} + p\_{-s+2})\left(1 - \frac{2}{s}\right)^2 + \dots,$$

where 0 <sup>≤</sup> *ps*, *<sup>p</sup>*−*s*, *ps*−1, *<sup>p</sup>*−*s*+1, *ps*−2, *<sup>p</sup>*−*s*+2,... <sup>≤</sup> 1 and <sup>∑</sup>*<sup>s</sup> <sup>α</sup>*0=−*<sup>s</sup> <sup>p</sup>α*<sup>0</sup> = 1. Therefore, we must have 0 ≤ *ξ* ≤ 1. In this case, equations (24) and (25) respectively take the forms

$$\eta\_2 = \frac{1}{2s^2} \left| \sin \theta\_1 + \cos \theta\_1 \right| \left[ \left\{ 3 \xi s^2 - s(s+1) \right\} \cos^2 \theta\_1 + \left\{ s(s+1) - \xi s^2 \right\} \right],\tag{26}$$

$$\begin{aligned} \{s(s+1) - \mathfrak{z}s^2\} \tan^3 \theta\_1 + \{7\mathfrak{z}s^2 - 3s(s+1)\} \tan^2 \theta\_1 + \\ \{8\mathfrak{z}s^2 - 2s(s+1)\} \tan \theta\_1 - 2\mathfrak{z}s^2 = 0. \end{aligned} \tag{27}$$

Let us first consider the input states of the form

$$\rho\_0 = p\_s |\vec{S} \cdot \mathfrak{A}\_0 \, s\rangle \langle \vec{S} \cdot \mathfrak{A}\_0 \, s| + p\_{-s} |\vec{S} \cdot \mathfrak{A}\_0 \, -s\rangle \langle \vec{S} \cdot \mathfrak{A}\_0 \, -s| \,\tag{28}$$

with *ps* + *<sup>p</sup>*−*<sup>s</sup>* = 1 and *ps*−1, *<sup>p</sup>*−*s*+1, *ps*−2, *<sup>p</sup>*−*s*+2,... = 0. Thus we see here that *<sup>ξ</sup>* = 1. Also equations (26) and (27) have respectively been turned into the forms

$$\eta\_2 = (1/2s)|\sin\theta\_1 + \cos\theta\_1|\{(2s-1)\cos^2\theta\_1 + 1\},\tag{29}$$

$$
\tan^3 \theta\_1 + (4s - 3)\tan^2 \theta\_1 + (6s - 2)\tan \theta\_1 - 2s = 0. \tag{30}
$$

As here *s* > 1, therefore the last equation will have only one positive root and the other two roots will be complex. The positive root will correspond to an angle *θmax* <sup>1</sup> (*s*) ∈ (0, *π*/4) for which it can be shown that *ηmax* <sup>2</sup> (*s*) <sup>≡</sup> *<sup>η</sup>*2(*θmax* <sup>1</sup> (*s*)) > 1 for all *s* > 1. Hence, in this case, BI is violated. If *ρ*<sup>0</sup> has contribution from neither of the states corresponding to *α*<sup>0</sup> = ±*s* (*i.e.*, *ps* = *p*−*<sup>s</sup>* = 0), we have

$$\xi = (p\_{s-1} + p\_{-s+1}) \left(1 - \frac{1}{s}\right)^2 + (p\_{s-2} + p\_{-s+2}) \left(1 - \frac{2}{s}\right)^2 + \dots \le \left(1 - \frac{1}{s}\right)^2 \dots$$

From equation (24) it follows that

$$\begin{aligned} \eta\_2 &= \frac{1}{\sqrt{2}s} \left| \sin \left( \theta\_1 + \pi/4 \right) \right| \left[ (s+1-s\tilde{\xi}) + (3s\tilde{\xi}-s-1) \cos^2 \theta\_1 \right] \\\\ &< \frac{1}{\sqrt{2}s} \times 1 \times \left[ (s+1-s\tilde{\xi}) + (3s\tilde{\xi}-s-1) \times 1 \right] = \sqrt{2}\tilde{\xi} \le \sqrt{2} \left( 1 - \frac{1}{s} \right)^2. \end{aligned}$$

But the quantity <sup>√</sup>2(<sup>1</sup> <sup>−</sup> 1/*s*)<sup>2</sup> is less than 1 for all *<sup>s</sup>* <sup>=</sup> 1/2, 1, 3/2, . . . , 6. Therefore, for *<sup>s</sup>* <sup>&</sup>gt; 1, if the initial state *ρ*<sup>0</sup> has contribution from neither of the states corresponding to *α*<sup>0</sup> = ±*s*, BI will be satisfied for all *s* ≤ 6. Thus we see that whenever *s* ∈ {3/2, 2, 5/2, . . . , 6}, in order that *ρ*<sup>0</sup> violates BI, the associated quantity *ξ* must have values near 1. In table 1, we have given the ranges of values of *ξ* (obtained numerically) for which BI is violated, starting from *s* = 1/2. The case when *s* → ∞ has also been considered in table 1.

Spin Measurements 13

Quantum Correlations in Successive Spin Measurements 209

<sup>2</sup> *<sup>f</sup>* <sup>&</sup>lt; 0.287 <sup>9</sup>

<sup>2</sup> *<sup>f</sup>* <sup>&</sup>lt; 0.254 <sup>11</sup>

1 *f* < 0.696 3 *f* < 0.267 5 *f* < 0.234

2 *f* < 0.321 4 *f* < 0.245 6 *f* < 0.227

compatible with the usual local HVT, is different in nature from the notion of classicality that

Table 3 answers the question, "what is the maximum fraction of noise that can be added to

<sup>0</sup> , which maximally breaks BI, so that the state has stronger than "classical correlations2?" We see that the corresponding fraction of noise (*i.e.*, for which BI is violated) decreases monotonically with *s*, or with the dimension of the Hilbert space. This may be compared with the results of Collins and Popescu (Collins & Popescu, 2001) who found that the nonlocal character of the correlations between the outcomes of measurements performed on entangled systems separated in space is robust in the presence of noise. They showed that, for any fraction of noise, by taking the Hilbert space of large enough dimension, one can find bipartite entangled states giving nonlocal correlations. These results have been obtained by considering two successive measurements on each part of the system. On the other hand, in the present case of successive measurements on the single spin state, we see that the fraction of noise that can be added so that the quantum correlations continue to break Bell inequality, falls off

<sup>2</sup> *f* < 0.239

<sup>2</sup> *f* < 0.230

∞ *f* < 0.195

<sup>16</sup> cos *<sup>θ</sup>*23{cos *<sup>θ</sup>*1[*<sup>M</sup>* cos<sup>2</sup> *<sup>θ</sup>*<sup>12</sup> <sup>+</sup> *<sup>N</sup>*] + *<sup>R</sup>*[3 cos2 *<sup>θ</sup>*<sup>12</sup> <sup>−</sup> <sup>1</sup>]} (35)

<sup>0</sup> + 1],

<sup>0</sup> + *s*(*s* + 1) − 3],

<sup>0</sup> − 3*s*(*s* + 1) + 1],

<sup>2</sup> all fractions *f* ≤ 1 are

<sup>2</sup> , BI is broken for all states. This indicates that the notion of "classicality",

*s f s f s f*

1

3

Table 3. The range of the noise *f* over which BI is violated.

would arise from the non-violation of BI here.

allowed, while for large s, *f* < 0.195.

**4.2 Three successive measurements (MKI)**

�*α*1*α*2*α*3� <sup>=</sup> <sup>1</sup>

shown that for *s* = <sup>1</sup>

*ρ*max

where

<sup>2</sup> <sup>0</sup> <sup>≤</sup> *<sup>f</sup>* <sup>≤</sup> <sup>1</sup> <sup>5</sup>

<sup>2</sup> *<sup>f</sup>* <sup>&</sup>lt; 0.395 <sup>7</sup>

monotonically with *s*, or the dimension of the Hilbert space. For *s* = <sup>1</sup>

*M* =

*N* =

*R* =

right-handed system (*a*ˆ1, *a*ˆ2,(*a*ˆ1 × *a*ˆ2)/|*a*ˆ1 × *a*ˆ2|)), etc.

+*s* ∑ *α*0=−*s*

+*s* ∑ *α*0=−*s*

+*s* ∑ *α*0=−*s*

<sup>2</sup> *i.e.*, correlations obeying "realism" and "locality in time", as described in section 3.

We again assume the input state to be given by equation (18). Using equation (A.19) :

*pα*0*α*0[9*α*<sup>2</sup>

*pα*0*α*0[5*α*<sup>2</sup>

*θ*<sup>1</sup> is the angle between *a*ˆ0 and *a*ˆ1 (measured with respect to the right-handed system (*a*ˆ0, *a*ˆ1,(*a*ˆ0 × *a*ˆ1)/|*a*ˆ0 × *a*ˆ1|)), *θ*<sup>12</sup> is the angle between *a*ˆ1 , *a*ˆ2 (measured with respect to the

*<sup>p</sup>α*0*α*0[5*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> <sup>3</sup>*α*<sup>2</sup>


Table 1. The ranges of *ξ*, for which BI is violated.


Table 2. The maximum violation of BI for different spin values for two successive measurements.

The maximum violation of Bell inequality, characterized by *η*2, decreases monotonically with *s*. Table 2 summarizes ( obtained numerically) the maximum allowed value of *η*<sup>2</sup> for each *s*. We see from this table that for all spin values *s*, BI is broken. Note that there is a sharp decrease in *η*<sup>2</sup> from *s* = <sup>1</sup> <sup>2</sup> to *s* = 1, while *η*<sup>2</sup> decreases slowly as *s* increases from 1. A possible reason is that, for *s* = 1/2, all states break BI while for *s* ≥ 1, only a fraction of spin states break it. We now consider a case where the initial state *ρmax* <sup>0</sup> (given in equation (28)) is contaminated by the maximally noisy state, resulting in the state

$$
\rho(f) = (1 - f)\,\rho\_0^{\text{max}} + \frac{f}{2s + 1}I\_\prime \tag{31}
$$

where the positive parameter *f* (≤ 1) is the probability of the noise contamination of the state *ρ*max <sup>0</sup> . Proceeding as before ( see equation (21)), we get

$$
\langle a\_1 a\_2 \rangle = \frac{1}{2} \cos \theta\_{12} [A' \cos^2 \theta\_1 + B'] \tag{32}
$$

where

$$A' = (1 - f)(2s - 1)s; \quad B' = (1 - f)s + \frac{2}{3}f(s + 1)s;$$

which leads to

$$\eta\_{\text{noise}} = \left(\frac{1}{2s^2}\right) (\sin\theta\_1 + \cos\theta\_1)(A'\cos^2\theta\_1 + B'). \tag{33}$$

Using the maximization procedure (*i.e.*, taking *∂ηnoise ∂θ*<sup>1</sup> = 0), tan*θ*<sup>1</sup> for maximum *ηnoise* is given by a real root of

$$B' \tan^3 \theta\_1 + (2A' - B') \tan^2 \theta\_1 + (3A' + B') \tan \theta\_1 - (A' + B') = 0. \tag{34}$$

The range of *f* for which *ηnoise* > 1 is tabulated in table 3. Note that for *s* = <sup>1</sup> <sup>2</sup> the state corresponding to *f* = 1 ( the random mixture) also breaks BI! Of course we have already


Table 3. The range of the noise *f* over which BI is violated.

shown that for *s* = <sup>1</sup> <sup>2</sup> , BI is broken for all states. This indicates that the notion of "classicality", compatible with the usual local HVT, is different in nature from the notion of classicality that would arise from the non-violation of BI here.

Table 3 answers the question, "what is the maximum fraction of noise that can be added to *ρ*max <sup>0</sup> , which maximally breaks BI, so that the state has stronger than "classical correlations2?" We see that the corresponding fraction of noise (*i.e.*, for which BI is violated) decreases monotonically with *s*, or with the dimension of the Hilbert space. This may be compared with the results of Collins and Popescu (Collins & Popescu, 2001) who found that the nonlocal character of the correlations between the outcomes of measurements performed on entangled systems separated in space is robust in the presence of noise. They showed that, for any fraction of noise, by taking the Hilbert space of large enough dimension, one can find bipartite entangled states giving nonlocal correlations. These results have been obtained by considering two successive measurements on each part of the system. On the other hand, in the present case of successive measurements on the single spin state, we see that the fraction of noise that can be added so that the quantum correlations continue to break Bell inequality, falls off monotonically with *s*, or the dimension of the Hilbert space. For *s* = <sup>1</sup> <sup>2</sup> all fractions *f* ≤ 1 are allowed, while for large s, *f* < 0.195.

#### **4.2 Three successive measurements (MKI)**

We again assume the input state to be given by equation (18). Using equation (A.19) :

$$\langle \mathfrak{a}\_1 \mathfrak{a}\_2 \mathfrak{a}\_3 \rangle = \frac{1}{16} \cos \theta\_{23} \{ \cos \theta\_1 [M \cos^2 \theta\_{12} + N] + R[3 \cos^2 \theta\_{12} - 1] \} \tag{35}$$

where

12 Will-be-set-by-IN-TECH

1 0 ≤ *ξ* ≤ 0.33 and 0.77 ≤ *ξ* ≤ 1 3 0.851 ≤ *ξ* ≤ 1 5 0.859 ≤ *ξ* ≤ 1

2 0.84 ≤ *ξ* ≤ 1 4 0.856 ≤ *ξ* ≤ 1 6 0.862 ≤ *ξ* ≤ 1

*s η*<sup>2</sup> *s η*<sup>2</sup> *s η*<sup>2</sup>

<sup>2</sup> 1.1638 <sup>9</sup>

<sup>2</sup> 1.1572 <sup>11</sup>

1 1.2112 3 1.1599 5 1.1526

2 1.17 4 1.1553 6 1.1509

The maximum violation of Bell inequality, characterized by *η*2, decreases monotonically with *s*. Table 2 summarizes ( obtained numerically) the maximum allowed value of *η*<sup>2</sup> for each *s*. We see from this table that for all spin values *s*, BI is broken. Note that there is a sharp decrease

where the positive parameter *f* (≤ 1) is the probability of the noise contamination of the state

that, for *s* = 1/2, all states break BI while for *s* ≥ 1, only a fraction of spin states break it.

*<sup>ρ</sup>*(*f*) = (<sup>1</sup> <sup>−</sup> *<sup>f</sup>*) *<sup>ρ</sup>*max

*A*� = (1 − *f*)(2*s* − 1)*s*; *B*� = (1 − *f*)*s* +

)tan2 *θ*<sup>1</sup> + (3*A*� + *B*�

corresponding to *f* = 1 ( the random mixture) also breaks BI! Of course we have already

The range of *f* for which *ηnoise* > 1 is tabulated in table 3. Note that for *s* = <sup>1</sup>

<sup>√</sup><sup>2</sup> <sup>5</sup>

Table 2. The maximum violation of BI for different spin values for two successive

<sup>2</sup> 1.1817 <sup>7</sup>

<sup>2</sup> 0.847 <sup>≤</sup> *<sup>ξ</sup>* <sup>≤</sup> <sup>1</sup> <sup>9</sup>

<sup>2</sup> 0.854 <sup>≤</sup> *<sup>ξ</sup>* <sup>≤</sup> <sup>1</sup> <sup>11</sup>

<sup>2</sup> 1.1538

<sup>2</sup> 1.1517

∞ 1.143

<sup>2</sup> to *s* = 1, while *η*<sup>2</sup> decreases slowly as *s* increases from 1. A possible reason is

<sup>0</sup> <sup>+</sup> *<sup>f</sup>*

<sup>2</sup> cos *<sup>θ</sup>*12[*A*� cos2 *<sup>θ</sup>*<sup>1</sup> <sup>+</sup> *<sup>B</sup>*�

(sin *θ*<sup>1</sup> + cos *θ*1)(*A*� cos2 *θ*<sup>1</sup> + *B*�

2

)tan *θ*<sup>1</sup> − (*A*� + *B*�

<sup>3</sup> *<sup>f</sup>*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>)*s*,

*∂θ*<sup>1</sup> = 0), tan*θ*<sup>1</sup> for maximum *ηnoise* is given

<sup>2</sup> 0.858 ≤ *ξ* ≤ 1

<sup>2</sup> 0.860 ≤ *ξ* ≤ 1

∞ 0.87 ≤ *ξ* ≤ 1

<sup>0</sup> (given in equation (28)) is contaminated

<sup>2</sup>*<sup>s</sup>* <sup>+</sup> <sup>1</sup> *<sup>I</sup>*, (31)

] (32)

). (33)

) = 0. (34)

<sup>2</sup> the state

*s ξ s ξ s ξ*

<sup>2</sup> *<sup>ξ</sup>* <sup>=</sup> <sup>1</sup> <sup>5</sup>

<sup>2</sup> 0.824 <sup>≤</sup> *<sup>ξ</sup>* <sup>≤</sup> <sup>1</sup> <sup>7</sup>

1 2

3

Table 1. The ranges of *ξ*, for which BI is violated.

We now consider a case where the initial state *ρmax*

<sup>0</sup> . Proceeding as before ( see equation (21)), we get

*ηnoise* =

Using the maximization procedure (*i.e.*, taking *∂ηnoise*

*<sup>B</sup>*� tan<sup>3</sup> *<sup>θ</sup>*<sup>1</sup> + (2*A*� <sup>−</sup> *<sup>B</sup>*�

�*α*1*α*2� <sup>=</sup> <sup>1</sup>

 1 2*s*<sup>2</sup> 

by the maximally noisy state, resulting in the state

1

3

measurements.

in *η*<sup>2</sup> from *s* = <sup>1</sup>

*ρ*max

where

which leads to

by a real root of

$$\begin{aligned} M &= \sum\_{\alpha\_0=-s}^{+s} p\_{\alpha\_0} \alpha\_0 [9a\_0^2 + s(s+1) - 3], \\ N &= \sum\_{\alpha\_0=-s}^{+s} p\_{\alpha\_0} \alpha\_0 [5s(s+1) - 3a\_0^2 + 1], \\ R &= \sum\_{\alpha\_0=-s}^{+s} p\_{\alpha\_0} \alpha\_0 [5a\_0^2 - 3s(s+1) + 1], \end{aligned}$$

*θ*<sup>1</sup> is the angle between *a*ˆ0 and *a*ˆ1 (measured with respect to the right-handed system (*a*ˆ0, *a*ˆ1,(*a*ˆ0 × *a*ˆ1)/|*a*ˆ0 × *a*ˆ1|)), *θ*<sup>12</sup> is the angle between *a*ˆ1 , *a*ˆ2 (measured with respect to the right-handed system (*a*ˆ1, *a*ˆ2,(*a*ˆ1 × *a*ˆ2)/|*a*ˆ1 × *a*ˆ2|)), etc.

<sup>2</sup> *i.e.*, correlations obeying "realism" and "locality in time", as described in section 3.

Spin Measurements 15

Quantum Correlations in Successive Spin Measurements 211

2

eigenvalues of *<sup>σ</sup><sup>z</sup>* are taken here as <sup>±</sup>1 instead of <sup>±</sup>(1/2). We also write <sup>|</sup>*αk*� for <sup>|</sup>

2

2*n*

where cos *<sup>θ</sup>k*−1,*<sup>k</sup>* = *<sup>a</sup>*ˆ*k*−<sup>1</sup> · *<sup>a</sup>*ˆ*<sup>k</sup>* for *<sup>k</sup>* = 1, 2, . . . , *<sup>n</sup>*. So, given the input state |*α*0�, the (joint) probability that the measurement outcomes will be *α*<sup>1</sup> ∈ {+1, −1} in the first measurement, *α*<sup>2</sup> ∈ {+1, −1} in the second measurement, . . ., *α<sup>n</sup>* ∈ {+1, −1} in the *n*-th measurement, will

> *n* ∏ *i*=1

Thus we see that given the input state *<sup>ρ</sup>*<sup>0</sup> = <sup>∑</sup>*α*0=±<sup>1</sup> *<sup>p</sup>α*<sup>0</sup> |*α*0��*α*0|, the average output state after

<sup>2</sup> particle in a mixed state. For simplicity we take the eigenvalues to be *α<sup>k</sup>* = ±1, *i.e.*, the

*<sup>ρ</sup>*<sup>0</sup> = *<sup>p</sup>*+1|*α*<sup>0</sup> = +1��*α*<sup>0</sup> = +1| + *<sup>p</sup>*−1|*α*<sup>0</sup> = −1��*α*<sup>0</sup> = −1|. (37)

*pα*<sup>0</sup> *p* (*α*1, *α*2,..., *αn*)|*αn*��*αn*| .

*αn*−1{coeff. of |*αn*−1��*αn*−1| in *ρn*−1}*α<sup>n</sup>* |�*αn*−1|*αn*�|

*<sup>α</sup>n*−1*αn*(<sup>1</sup> + *<sup>α</sup>i*−1*α<sup>i</sup>* cos *<sup>θ</sup>i*−1,*i*)

*<sup>α</sup>n*(<sup>1</sup> + *<sup>α</sup>i*−1*α<sup>i</sup>* cos *<sup>θ</sup>i*−1,*i*)

*<sup>α</sup>n*−1*α<sup>n</sup> <sup>p</sup>*(*α*1, *<sup>α</sup>*2, ··· , *<sup>α</sup>n*)

= cos *<sup>θ</sup>n*−1,*<sup>n</sup>* (40)

*α<sup>n</sup> p* (*α*1, *α*2,..., *αn*)

= (*p*+<sup>1</sup> − *<sup>p</sup>*−1) cos *<sup>θ</sup>*<sup>1</sup> cos *<sup>θ</sup>*<sup>12</sup> ··· cos *<sup>θ</sup>n*−1,*<sup>n</sup>* (41)

*n* ∏ *i*=1

*n* ∏ *i*=1 *S* · *a*ˆ*i*,(*i* = 1, 2, 3, . . . , *n*) on a spin

(<sup>1</sup> + *<sup>α</sup>k*−1*α<sup>k</sup>* cos *<sup>θ</sup>k*−1,*k*) (38)

(<sup>1</sup> + *<sup>α</sup>i*−1*α<sup>i</sup>* cos *<sup>θ</sup>i*−1,*i*). (39)

*S* · *a*ˆ*k*, *αk*�.

2

**5.** *n* **successive measurements for Spin-** <sup>1</sup>

**5.1 Violation Mermin-Klyshko Inequality (MKI)**

<sup>2</sup> system, we have

*n* successive measurements will be given by

�*αn*−1*αn*�*QM* <sup>=</sup> <sup>∑</sup> *<sup>α</sup>n*−1,*αn*=±<sup>1</sup>

by equation (39). Further

= ∑ *α*0=±1

= ∑ *α*0=±1

�*αn*�*QM* = ∑

*αn*=±1

= ∑ *α*0=±1

= ∑ *α*0=±1

*s* = <sup>1</sup>

For a spin- <sup>1</sup>

be given by

The initial state is taken as

We consider now *n* successive measurements in direction


*<sup>p</sup>*(*α*1, *<sup>α</sup>*2, ··· , *<sup>α</sup>n*) = <sup>1</sup>

*α*0,*α*1,...,*αn*=±1

*pα*<sup>0</sup> ∑

*α*1,*α*2,...,*αn*=±1

*α*1,*α*2,...,*αn*=±1

*α<sup>n</sup>* {coeff. of |*αn*��*αn*| in *ρn*}

*α*1,*α*2,...,*αn*=±1

*<sup>p</sup>α*<sup>0</sup> <sup>2</sup>−*<sup>n</sup>* ∑

*pα*<sup>0</sup> ∑

*α*1,*α*2,...,*αn*=±1

*<sup>p</sup>α*<sup>0</sup> <sup>2</sup>−*<sup>n</sup>* ∑

*ρ<sup>n</sup>* = ∑

Then, for *n* successive measurements on spin-1/2 system,


Table 4. The maximum violation of MKI for different spin values for three successive measurements.

We now consider the pure state <sup>|</sup>� *<sup>S</sup>* · *<sup>a</sup>*ˆ0,*s*��� *S* · *a*ˆ0,*s*| instead of considering the most general state *<sup>ρ</sup>*0, given in equation (18). So here *<sup>M</sup>* <sup>=</sup> *<sup>s</sup>*(2*<sup>s</sup>* <sup>−</sup> <sup>1</sup>)(5*<sup>s</sup>* <sup>+</sup> <sup>3</sup>), *<sup>N</sup>* <sup>=</sup> *<sup>s</sup>*(2*s*<sup>2</sup> <sup>+</sup> <sup>5</sup>*<sup>s</sup>* <sup>+</sup> <sup>1</sup>), and *<sup>R</sup>* <sup>=</sup> *<sup>s</sup>*(2*<sup>s</sup>* <sup>−</sup> 1)(*s* − 1). Substituting the correlations like that in equation (35) in the MKI (given in equation (9)), using the above-mentioned values of *M*, *N*, *R*, and then finding out the conditions ( numerically) for which *<sup>η</sup>*<sup>3</sup> ≡ |*MKI*|/*s*<sup>3</sup> is maximized, we get the maximum possible *<sup>η</sup>*3-values for different spins as summarized in table 4.

We see that *η*<sup>3</sup> > 1 for all spins and *η*<sup>3</sup> > *η*<sup>2</sup> except *s* = <sup>1</sup> <sup>2</sup> , while *<sup>η</sup>*<sup>3</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>√</sup>2 for *<sup>s</sup>* <sup>=</sup> 1/2. Also *η*3, like *η*2, decreases monotonically with *s*. It is interesting, in the case of two and three successive measurements of spin *s* prepared in a pure state, the maximum violation of BI and MKI tends to a constant for arbitrary large *s* .

$$
\eta\_3(s \to \infty) = 1.153 \text{ (approx.)}.
$$

$$
\eta\_2(s \to \infty) = 1.143 \text{ (approx.)}.
$$

It is thus seen that large quantum numbers do not guarantee "classical" ( as defined in this chapter) behavior.

It is straightforward to check that, three successive measurements satisfy Svetlichny Inequality (SI) (equation (10)). The reason is that, for all *s*, the settings of the measurement directions which maximize *MKI*� are obtained from those which maximize *MKI* by interchanging primes on the corresponding unit vectors. Thus these two settings are incompatible so that we cannot get a single set of measurement directions, which maximize both *MKI* and *MKI*� . In fact, for all *s*, the measurement directions which maximize *MKI* (*MKI*� ) correspond to *MKI*� = 0 (*MKI* = 0).

We now consider the situation of three consecutive observations but two-fold correlations for two measurements � *S*.*a*ˆ1 and � *S*.*a*ˆ3 performed, say, at time *t*<sup>1</sup> and *t*3, but where an additional measurement (� *S*.*a*ˆ2) is performed at time *t*<sup>2</sup> lying between *t*<sup>1</sup> and *t*<sup>3</sup> (*t*<sup>1</sup> < *t*<sup>2</sup> < *t*3).

By substituting Eq (A.20) in Bell type inequality Eq(8) and simplifying, we obtain:

$$\begin{split} |BI| &= \frac{1}{2} |[\cos\theta\_{32} + \cos\theta\_{32}] \langle a\_1 a\_2 \rangle + [\cos\theta\_{32} - \cos\theta\_{32}] \langle a\_1' a\_2 \rangle | \\ &\le \frac{1}{2} |[\cos\theta\_{32} + \cos\theta\_{32}] ||\langle a\_1 a\_2 \rangle| + |[\cos\theta\_{32} - \cos\theta\_{32}] ||\langle a\_1' a\_2 \rangle| \\ &\le \cos\theta\_{32} s^2 \le s^2. \end{split} \tag{36}$$

We have used *max*|�*α*1*α*2�| = *max*|�*α*� <sup>1</sup>*α*2�| <sup>=</sup> *<sup>s</sup>*2.

So, the correlation function (36) for a given measurement performed at *t*<sup>2</sup> cannot violate the Bell type inequality for measurements at *t*<sup>1</sup> and *t*3. Therefore, any measurement performed at time *t*<sup>2</sup> "disentangles" events at time *t*<sup>1</sup> and *t*<sup>3</sup> if *t*<sup>1</sup> < *t*<sup>2</sup> < *t*<sup>3</sup> (Brukner et al., 2004).

#### **5.** *n* **successive measurements for Spin-** <sup>1</sup> 2

#### **5.1 Violation Mermin-Klyshko Inequality (MKI)**

We consider now *n* successive measurements in direction *S* · *a*ˆ*i*,(*i* = 1, 2, 3, . . . , *n*) on a spin *s* = <sup>1</sup> <sup>2</sup> particle in a mixed state. For simplicity we take the eigenvalues to be *α<sup>k</sup>* = ±1, *i.e.*, the eigenvalues of *<sup>σ</sup><sup>z</sup>* are taken here as <sup>±</sup>1 instead of <sup>±</sup>(1/2). We also write <sup>|</sup>*αk*� for <sup>|</sup> *S* · *a*ˆ*k*, *αk*�. The initial state is taken as

$$p\_0 = p\_{+1}|a\_0 = +1\rangle\langle a\_0 = +1| + p\_{-1}|a\_0 = -1\rangle\langle a\_0 = -1|.\tag{37}$$

For a spin- <sup>1</sup> <sup>2</sup> system, we have

14 Will-be-set-by-IN-TECH

*s η*<sup>3</sup> *s η*<sup>3</sup> *s η*<sup>3</sup>

<sup>2</sup> 1.1736 <sup>9</sup>

<sup>2</sup> 1.1670 <sup>11</sup>

1 1.2178 3 1.1698 5 1.1621

2 1.1793 4 1.1650 6 1.1601

*<sup>ρ</sup>*0, given in equation (18). So here *<sup>M</sup>* <sup>=</sup> *<sup>s</sup>*(2*<sup>s</sup>* <sup>−</sup> <sup>1</sup>)(5*<sup>s</sup>* <sup>+</sup> <sup>3</sup>), *<sup>N</sup>* <sup>=</sup> *<sup>s</sup>*(2*s*<sup>2</sup> <sup>+</sup> <sup>5</sup>*<sup>s</sup>* <sup>+</sup> <sup>1</sup>), and *<sup>R</sup>* <sup>=</sup> *<sup>s</sup>*(2*<sup>s</sup>* <sup>−</sup> 1)(*s* − 1). Substituting the correlations like that in equation (35) in the MKI (given in equation (9)), using the above-mentioned values of *M*, *N*, *R*, and then finding out the conditions ( numerically) for which *<sup>η</sup>*<sup>3</sup> ≡ |*MKI*|/*s*<sup>3</sup> is maximized, we get the maximum possible *<sup>η</sup>*3-values

Also *η*3, like *η*2, decreases monotonically with *s*. It is interesting, in the case of two and three successive measurements of spin *s* prepared in a pure state, the maximum violation of BI and

*η*3(*s* → ∞) = 1.153 (approx.),

*η*2(*s* → ∞) = 1.143 (approx.). It is thus seen that large quantum numbers do not guarantee "classical" ( as defined in this

It is straightforward to check that, three successive measurements satisfy Svetlichny Inequality (SI) (equation (10)). The reason is that, for all *s*, the settings of the measurement

interchanging primes on the corresponding unit vectors. Thus these two settings are incompatible so that we cannot get a single set of measurement directions, which maximize

We now consider the situation of three consecutive observations but two-fold correlations for

By substituting Eq (A.20) in Bell type inequality Eq(8) and simplifying, we obtain:

<sup>1</sup>*α*2�| <sup>=</sup> *<sup>s</sup>*2.

time *t*<sup>2</sup> "disentangles" events at time *t*<sup>1</sup> and *t*<sup>3</sup> if *t*<sup>1</sup> < *t*<sup>2</sup> < *t*<sup>3</sup> (Brukner et al., 2004).

So, the correlation function (36) for a given measurement performed at *t*<sup>2</sup> cannot violate the Bell type inequality for measurements at *t*<sup>1</sup> and *t*3. Therefore, any measurement performed at

*S*.*a*ˆ2) is performed at time *t*<sup>2</sup> lying between *t*<sup>1</sup> and *t*<sup>3</sup> (*t*<sup>1</sup> < *t*<sup>2</sup> < *t*3).

<sup>2</sup>]�*α*1*α*2� + [cos *θ*<sup>32</sup> − cos *θ*3�

<sup>2</sup>]||�*α*1*α*2�| + |[cos *θ*<sup>32</sup> − cos *θ*3�

Table 4. The maximum violation of MKI for different spin values for three successive

<sup>2</sup> 1.1634

<sup>2</sup> 1.1610

∞ 1.1527

*S* · *a*ˆ0,*s*| instead of considering the most general state

are obtained from those which maximize *MKI* by

2]�*α*� <sup>1</sup>*α*2�|

2]||�*α*�

<sup>1</sup>*α*2�|

. In fact, for all *s*, the measurement directions which maximize *MKI*

*S*.*a*ˆ3 performed, say, at time *t*<sup>1</sup> and *t*3, but where an additional

2. (36)

<sup>2</sup> , while *<sup>η</sup>*<sup>3</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>√</sup>2 for *<sup>s</sup>* <sup>=</sup> 1/2.

<sup>√</sup><sup>2</sup> <sup>5</sup>

<sup>2</sup> 1.1907 <sup>7</sup>

*<sup>S</sup>* · *<sup>a</sup>*ˆ0,*s*���

1 2

3

measurements.

chapter) behavior.

both *MKI* and *MKI*�

two measurements �

measurement (�

(*MKI*�

We now consider the pure state <sup>|</sup>�

for different spins as summarized in table 4.

MKI tends to a constant for arbitrary large *s* .

directions which maximize *MKI*�

<sup>|</sup>*BI*<sup>|</sup> <sup>=</sup> <sup>1</sup> 2

> ≤ 1 2

We have used *max*|�*α*1*α*2�| = *max*|�*α*�

≤ cos *θ*32*s*

) correspond to *MKI*� = 0 (*MKI* = 0).

*S*.*a*ˆ1 and �



<sup>2</sup> <sup>≤</sup> *<sup>s</sup>*

We see that *η*<sup>3</sup> > 1 for all spins and *η*<sup>3</sup> > *η*<sup>2</sup> except *s* = <sup>1</sup>

$$\left| \left< a\_{k-1} | a\_k \right> \right|^2 = \frac{1}{2} (1 + a\_{k-1} a\_k \cos \theta\_{k-1,k}) \tag{38}$$

where cos *<sup>θ</sup>k*−1,*<sup>k</sup>* = *<sup>a</sup>*ˆ*k*−<sup>1</sup> · *<sup>a</sup>*ˆ*<sup>k</sup>* for *<sup>k</sup>* = 1, 2, . . . , *<sup>n</sup>*. So, given the input state |*α*0�, the (joint) probability that the measurement outcomes will be *α*<sup>1</sup> ∈ {+1, −1} in the first measurement, *α*<sup>2</sup> ∈ {+1, −1} in the second measurement, . . ., *α<sup>n</sup>* ∈ {+1, −1} in the *n*-th measurement, will be given by

$$p(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) = \frac{1}{2^n} \prod\_{i=1}^n (1 + \mathfrak{a}\_{i-1} \mathfrak{a}\_i \cos \theta\_{i-1,i}).\tag{39}$$

Thus we see that given the input state *<sup>ρ</sup>*<sup>0</sup> = <sup>∑</sup>*α*0=±<sup>1</sup> *<sup>p</sup>α*<sup>0</sup> |*α*0��*α*0|, the average output state after *n* successive measurements will be given by

$$\rho\_{\mathfrak{n}} = \sum\_{\mathfrak{a}\_{\mathfrak{d}}, \mathfrak{a}\_{\mathfrak{1}}, \dots, \mathfrak{a}\_{\mathfrak{n}} = \pm 1} p\_{\mathfrak{a}\_{\mathfrak{l}}} p \left( \mathfrak{a}\_{\mathfrak{1}}, \mathfrak{a}\_{\mathfrak{2}}, \dots, \mathfrak{a}\_{\mathfrak{n}} \right) \left| \mathfrak{a}\_{\mathfrak{n}} \right\rangle \left\langle \mathfrak{a}\_{\mathfrak{n}} \right| \dots$$

Then, for *n* successive measurements on spin-1/2 system,

$$\begin{split} \langle a\_{n-1} a\_{n} \rangle\_{QM} &= \sum\_{a\_{n-1}, a\_{n}=\pm 1} a\_{n-1} \{ \text{coeff. of } |a\_{n-1}\rangle \langle a\_{n-1}| \text{ in } \rho\_{n-1} \} a\_{n} \left| \langle a\_{n-1} | a\_{n} \rangle \right|^{2} \\ &= \sum\_{a\_{0}=\pm 1} p\_{a\_{0}} \sum\_{a\_{1}, a\_{2}, \dots, a\_{n}=\pm 1} a\_{n-1} a\_{n} p(a\_{1}, a\_{2}, \dots, a\_{n}) \\ &= \sum\_{a\_{0}=\pm 1} p\_{a\_{0}} 2^{-n} \sum\_{a\_{1}, a\_{2}, \dots, a\_{n}=\pm 1} \prod\_{i=1}^{n} a\_{n-1} a\_{i} (1 + a\_{i-1} a\_{i} \cos \theta\_{i-1,i}) \\ &= \cos \theta\_{n-1,n} \end{split} \tag{40}$$

by equation (39). Further

$$\begin{aligned} \langle a\_{\boldsymbol{n}} \rangle\_{QM} &= \sum\_{a\_{\boldsymbol{n}}= \pm 1} a\_{\boldsymbol{n}} \left\{ \text{coeff. of } |a\_{\boldsymbol{n}}\rangle \langle a\_{\boldsymbol{n}}| \text{ in } \rho\_{\boldsymbol{n}} \right\} \\ &= \sum\_{a\_{\boldsymbol{0}}= \pm 1} p\_{\boldsymbol{n}\_{0}} \sum\_{a\_{1}, a\_{2}, \dots, a\_{\boldsymbol{n}}= \pm 1} a\_{\boldsymbol{n}} p \left( a\_{1}, a\_{2}, \dots, a\_{\boldsymbol{n}} \right) \\ &= \sum\_{a\_{\boldsymbol{0}}= \pm 1} p\_{\boldsymbol{n}\_{0}} 2^{-n} \sum\_{a\_{1}, a\_{2}, \dots, a\_{\boldsymbol{n}}= \pm 1} \prod\_{i=1}^{n} a\_{\boldsymbol{n}} (1 + a\_{i-1} a\_{i} \cos \theta\_{i-1, i}) \\ &= (p\_{+1} - p\_{-1}) \cos \theta\_{1} \cos \theta\_{12} \cdots \cos \theta\_{n-1, n} \end{aligned} \tag{41}$$

Spin Measurements 17

Quantum Correlations in Successive Spin Measurements 213

*<sup>k</sup>*−1, *<sup>φ</sup>*�

*<sup>k</sup>* cos(*φk*−<sup>1</sup> − *<sup>φ</sup>*�

*<sup>k</sup>* respectively, where all *θ* ∈ [0, *π*] and all *φ* ∈ [0, 2*π*]. Then

*<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>φ</sup>k*)]

*<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>φ</sup>*�

= <sup>0</sup> ⇒ *<sup>φ</sup>k*−<sup>1</sup> = *<sup>φ</sup>*�

*<sup>k</sup>* + sin *<sup>θ</sup>k*−<sup>1</sup> sin *<sup>θ</sup>*�

*<sup>k</sup>* cos(*φ*�

*<sup>k</sup>*) + cos(*θ*�

*<sup>k</sup>*) + cos(*θ*�

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> 0,

*<sup>k</sup>*−2�|} ≤ <sup>√</sup>

*QM* (*i*, *j*)] the maximal value of the Bell expression for qubits *i* and

*<sup>k</sup>*−<sup>1</sup> sin *<sup>θ</sup><sup>k</sup>* cos(*φ*�

*<sup>k</sup>*−2�[cos *<sup>θ</sup>k*−<sup>1</sup> cos *<sup>θ</sup><sup>k</sup>* <sup>+</sup> sin *<sup>θ</sup>k*−<sup>1</sup> sin *<sup>θ</sup><sup>k</sup>* cos(*φk*−<sup>1</sup> <sup>−</sup> *<sup>φ</sup>k*)

*<sup>k</sup>*−<sup>1</sup> sin *<sup>θ</sup>*�

<sup>=</sup> *<sup>∂</sup>*|�*Mk*�| *∂φ*� *k*

*<sup>k</sup>*−2�[cos(*θk*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>k*) <sup>−</sup> (cos *<sup>θ</sup>*�

*<sup>k</sup>*−2�|[cos(*θk*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>k*) <sup>−</sup> (cos *<sup>θ</sup>*�

*<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>k*, *<sup>z</sup>* <sup>=</sup> *<sup>θ</sup>k*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup><sup>k</sup>* and *<sup>θ</sup>*�

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> *<sup>∂</sup>*|�*Mk*�|

*<sup>k</sup>*−1), (*θk*, *<sup>φ</sup>k*), (*θ*�

*k*)

*<sup>k</sup>*)]|. (47)

*k*.

*<sup>k</sup>*)]|. (48)

*<sup>k</sup>* = *x* + *y* − *z* in

*<sup>k</sup>* lie on the same

*<sup>k</sup>*−<sup>1</sup> <sup>=</sup> *<sup>φ</sup><sup>k</sup>* <sup>=</sup> *<sup>φ</sup>*�

*<sup>k</sup>*−1, *<sup>a</sup>*ˆ*k*, *<sup>a</sup>*ˆ�

*<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*�

*<sup>k</sup>*�| ≤ 2, we obtain:

2. (50)

QM (2, 3)] ≤ 2. (51)

2. (49)

*<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>k*)]

*<sup>k</sup>*)]|

*<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>k*)]

*<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*�

*<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*�

*<sup>k</sup>*, *φ*� *k*)

We now consider the spherical-polar co-ordinates (*θk*−1, *<sup>φ</sup>k*−1), (*θ*�


*<sup>k</sup>*−<sup>1</sup> cos *<sup>θ</sup><sup>k</sup>* <sup>+</sup> sin *<sup>θ</sup>*�

*<sup>k</sup>* − sin *θ*�



*<sup>∂</sup><sup>x</sup>* <sup>=</sup> *<sup>∂</sup>*|�*Mk*�|

<sup>2</sup> {|�*Mk*−2� <sup>+</sup> �*M*�

*<sup>η</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup>

One can get this result by induction hypothesis. Thus, we conclude that QM violates the MKI

Although in contrast to correlations in space there are no genuine multi-mode correlations in time, we will see that temporal correlations can be stronger than spatial ones in a certain

*j* (Bell inequality is obtained by Eq(8)). Scarani and Gisin (Scarani & Gisin, 2001) found an

QM (1, 2)] + max[Bspace

*<sup>k</sup>*−<sup>1</sup> cos *<sup>θ</sup>*�

<sup>=</sup> *<sup>∂</sup>*|�*Mk*�| *∂φ*� *k*−1

The maximum value of |�*Mk*�| will occur when all the vectors *<sup>a</sup>*ˆ*k*−1, *<sup>a</sup>*ˆ�

*<sup>k</sup>*, *y* = *θ*�

*∂*|�*Mk*�|

we get *x* = *y* = −*z* = *π*/4. Finally, by using the fact |�*Mk*�| + |�*M*�

√2


(|�*Mn*�| ≤ 1) for *<sup>n</sup>* successive measurements upto <sup>√</sup>2, *i.e.*,

interesting bound that holds for arbitrary state of three qubits: max[Bspace

**5.2 Violation Scarani-Gisin inequality (SCI)**

sense. We denote by max[*Bspace*

*<sup>k</sup>*−1, *<sup>a</sup>*ˆ*k*, *<sup>a</sup>*ˆ�

2

+ cos *θ*�

+ �*M*�

− cos *θ*�

<sup>=</sup> *<sup>∂</sup>*|�*Mk*�| *∂φ<sup>k</sup>*


2

+ �*M*�

≤ 1 2

+ 1 2 |�*M*�


of the vectors *<sup>a</sup>*ˆ*k*−1, *<sup>a</sup>*ˆ�


We know from

plane. We obtain:

*∂*|�*Mk*�| *∂φk*−<sup>1</sup>

By substituting *<sup>x</sup>* = *<sup>θ</sup>k*−<sup>1</sup> − *<sup>θ</sup>*�

above-equation and by using

where *θ*<sup>1</sup> ≡ *θ*0,1, *θ*<sup>12</sup> ≡ *θ*1,2, etc. Now equations (40) and (41) give,

$$
\langle \mathfrak{a}\_{\mathfrak{n}} \rangle\_{QM} = \langle \mathfrak{a}\_{1} \rangle\_{QM} \langle \mathfrak{a}\_{2} \mathfrak{a}\_{3} \rangle\_{QM} \cdots \times \langle \mathfrak{a}\_{n-1} \mathfrak{a}\_{n} \rangle\_{QM}.\tag{42}
$$

Further,

$$\begin{aligned} \langle a\_{n-k} \cdots a\_n \rangle\_{QM} &= \\ \sum\_{a\_0} p\_{a\_0} 2^{-n} \sum\_{\substack{\mathfrak{a}\_1, \mathfrak{a}\_2, \dots \mathfrak{a}\_n = \pm 1}} \prod\_{i=1}^n (\mathfrak{a}\_{n-k} \cdots \mathfrak{a}\_n)(1 + \mathfrak{a}\_{i-1} \mathfrak{a}\_i \cos \theta\_{i-1,i}) &= \\ \left\{ \begin{array}{l} \langle \mathfrak{a}\_1 \rangle\_{QM} \langle \mathfrak{a}\_2 \mathfrak{a}\_3 \rangle\_{QM} \cdots \langle \mathfrak{a}\_{n-1} \mathfrak{a}\_n \rangle\_{QM} & k \text{ even} \\\\ \langle \mathfrak{a}\_{n-k} \mathfrak{a}\_{n-k+1} \rangle\_{QM} \langle \mathfrak{a}\_{n-k+2} \mathfrak{a}\_{n-k+3} \rangle\_{QM} \cdots \langle \mathfrak{a}\_{n-1} \mathfrak{a}\_n \rangle\_{QM} & k \text{ odd} \end{cases} \end{aligned} \right. (10.107)$$

All of the above results are inherently quantum and are not compatible with HVT ( see the discussion in the next paragraph). The first two results ((41) and (42)) are the special cases of the last result (43) for *k* = 1 and *k* = 0 (with *α*<sup>0</sup> = 1). If the number of variables ( which are averaged) is odd (i.e. *k* is even) the average depends on the measurements prior to (*n* − *k*), while in the other case the average does not depend on the measurements prior to (*n* − *k*). For example, for two successive measurements ( taking *n* = 2 and *k* = 1), gives �*α*1*α*2� = cos *θ*12, which is independent of the initial state. On the other hand, for three successive measurements ( taking *n* = 3 and *k* = 2), we have �*α*1*α*2*α*3� = �*α*1��*α*2*α*3� – showing its dependence on the initial state (as �*α*1� = (*p*+<sup>1</sup> − *<sup>p</sup>*−1)cos*θ*<sup>1</sup> depends upon the initial state *<sup>ρ</sup>*<sup>0</sup> = <sup>∑</sup>*α*0=±<sup>1</sup> *<sup>p</sup>α*<sup>0</sup> |*α*0��*α*0|). Moreover, the correlation �*α*1*α*2*α*3*α*4�*QM* for four successive measurements ( for example ) turns out to be dependent only on the two 'disjoint' correlations �*α*1*α*2�*QM* and �*α*3*α*4�*QM* . In general, we have:

$$
\langle \mathfrak{a}\_1 \mathfrak{a}\_2, \dots, \mathfrak{a}\_{2p} \rangle = \langle \mathfrak{a}\_1 \mathfrak{a}\_2 \rangle \langle \mathfrak{a}\_3 \mathfrak{a}\_4 \rangle \dots \langle \mathfrak{a}\_{2p-1} \mathfrak{a}\_{2p} \rangle \tag{44}
$$

and

$$
\langle \mathfrak{a}\_1 \mathfrak{a}\_2, \dots, \mathfrak{a}\_{2p+1} \rangle = \langle \mathfrak{a}\_1 \rangle \langle \mathfrak{a}\_2 \mathfrak{a}\_3 \rangle \dots \langle \mathfrak{a}\_{2p} \mathfrak{a}\_{2p+1} \rangle . \tag{45}
$$

Interestingly if *a*ˆ0 ⊥ *a*ˆ1 so that cos*θ*<sup>1</sup> = 0 ( and so, �*α*1�*QM* = 0) or, if the initial state is the random mixture (1/2) <sup>∑</sup>*α*0=±<sup>1</sup> |*α*0��*α*0| ( and so �*α*1� = 0), then for all even *<sup>k</sup>*,

$$\langle \alpha\_{n-k} \cdot \cdot \cdot \alpha\_n \rangle\_{QM} = 0$$

and so

$$
\langle \alpha\_1 \alpha\_2 \cdot \cdot \cdot \alpha\_{n=2p+1} \rangle\_{QM} = 0.
$$

We shall now show that for *n* successive experiments ( with *n* > 1), QM violates the inequality |�*MKI*�| ≤ *<sup>s</sup>*<sup>3</sup> ( see equation (9)) up to <sup>√</sup>2 for *<sup>s</sup>* <sup>=</sup> 1/2 systems. We take the eigenvalues to be *α<sup>k</sup>* = ±1 so |�*Mk*�|*HVT* ≤ 1). We have already shown that for *n* = 2 and *n* = 3, the corresponding MKI's are violated (section 4). Now, we know that temporal two-fold correlations �*αk*−1*α*� *<sup>k</sup>*� , �*α*� *<sup>k</sup>*−1*αk*�, �*αk*−1*αk*� and �*α*� *<sup>k</sup>*−1*α*� *<sup>k</sup>*� are independent on the previous measurements. So by using equations (43) and (13) we find that

$$|\langle M\_k \rangle| = \frac{1}{2} |\langle M\_{k-2} \rangle| \left[ \langle a\_{k-1} a\_k' \rangle + \langle a\_{k-1}' a\_k \rangle \right] + \langle M\_{k-2}' \rangle \left[ \langle a\_{k-1} a\_k \rangle - \langle a\_{k-1}' a\_k' \rangle \right]|. \tag{46}$$

We now consider the spherical-polar co-ordinates (*θk*−1, *<sup>φ</sup>k*−1), (*θ*� *<sup>k</sup>*−1, *<sup>φ</sup>*� *<sup>k</sup>*−1), (*θk*, *<sup>φ</sup>k*), (*θ*� *<sup>k</sup>*, *φ*� *k*) of the vectors *<sup>a</sup>*ˆ*k*−1, *<sup>a</sup>*ˆ� *<sup>k</sup>*−1, *<sup>a</sup>*ˆ*k*, *<sup>a</sup>*ˆ� *<sup>k</sup>* respectively, where all *θ* ∈ [0, *π*] and all *φ* ∈ [0, 2*π*]. Then |�*Mk*�| has the form

$$\begin{split} |\langle M\_{k}\rangle| &= \frac{1}{2} |\langle M\_{k-2}\rangle| \cos\theta\_{k-1} \cos\theta\_{k}^{\prime} + \sin\theta\_{k-1} \sin\theta\_{k}^{\prime} \cos(\phi\_{k-1} - \phi\_{k}^{\prime}) \\ &+ \cos\theta\_{k-1}^{\prime} \cos\theta\_{k} + \sin\theta\_{k-1}^{\prime} \sin\theta\_{k} \cos(\phi\_{k-1}^{\prime} - \phi\_{k}) ] \\ &+ \langle M\_{k-2}^{\prime} \rangle [\cos\theta\_{k-1} \cos\theta\_{k} + \sin\theta\_{k-1} \sin\theta\_{k} \cos(\phi\_{k-1} - \phi\_{k}) \\ &- \cos\theta\_{k-1}^{\prime} \cos\theta\_{k}^{\prime} - \sin\theta\_{k-1}^{\prime} \sin\theta\_{k}^{\prime} \cos(\phi\_{k-1}^{\prime} - \phi\_{k}^{\prime})]. \end{split} \tag{47}$$

We know from

16 Will-be-set-by-IN-TECH

�*αn*−*<sup>k</sup>* ··· *<sup>α</sup>n*�*QM* =

�*α*1�*QM*�*α*2*α*3�*QM* ···�*αn*−1*αn*�*QM k* even

�*αn*−*kαn*−*k*+1�*QM*�*αn*−*k*+2*αn*−*k*+3�*QM* ···�*αn*−1*αn*�*QM <sup>k</sup>* odd

All of the above results are inherently quantum and are not compatible with HVT ( see the discussion in the next paragraph). The first two results ((41) and (42)) are the special cases of the last result (43) for *k* = 1 and *k* = 0 (with *α*<sup>0</sup> = 1). If the number of variables ( which are averaged) is odd (i.e. *k* is even) the average depends on the measurements prior to (*n* − *k*), while in the other case the average does not depend on the measurements prior to (*n* − *k*). For example, for two successive measurements ( taking *n* = 2 and *k* = 1), gives �*α*1*α*2� = cos *θ*12, which is independent of the initial state. On the other hand, for three successive measurements ( taking *n* = 3 and *k* = 2), we have �*α*1*α*2*α*3� = �*α*1��*α*2*α*3� – showing its dependence on the initial state (as �*α*1� = (*p*+<sup>1</sup> − *<sup>p</sup>*−1)cos*θ*<sup>1</sup> depends upon the initial state *<sup>ρ</sup>*<sup>0</sup> = <sup>∑</sup>*α*0=±<sup>1</sup> *<sup>p</sup>α*<sup>0</sup> |*α*0��*α*0|). Moreover, the correlation �*α*1*α*2*α*3*α*4�*QM* for four successive measurements ( for example ) turns out to be dependent only on the two 'disjoint'

∏*<sup>n</sup>*

�*αn*�*QM* = �*α*1�*QM*�*α*2*α*3�*QM* ···�*αn*−1*αn*�*QM*. (42)

*<sup>i</sup>*=1(*αn*−*<sup>k</sup>* ··· *<sup>α</sup>n*)(<sup>1</sup> + *<sup>α</sup>i*−1*α<sup>i</sup>* cos *<sup>θ</sup>i*−1,*i*) =

= �*α*1*α*2��*α*3*α*4�...�*α*2*p*−1*α*2*p*� (44)

= �*α*1��*α*2*α*3�...�*α*2*pα*2*p*+1�. (45)

*<sup>k</sup>*−2�[�*αk*−1*αk*�−�*α*�

*<sup>k</sup>*� are independent on the previous

*<sup>k</sup>*−1*α*� *<sup>k</sup>*�]|.

(46)

(43)

where *θ*<sup>1</sup> ≡ *θ*0,1, *θ*<sup>12</sup> ≡ *θ*1,2, etc. Now equations (40) and (41) give,

*<sup>p</sup>α*<sup>0</sup> <sup>2</sup>−*<sup>n</sup>* <sup>∑</sup> *<sup>α</sup>*1,*α*2,...,*αn*=±<sup>1</sup>

correlations �*α*1*α*2�*QM* and �*α*3*α*4�*QM* . In general, we have:

*α*1*α*2,..., *α*2*<sup>p</sup>*

*α*1*α*2,..., *α*2*p*+<sup>1</sup>

�

�

random mixture (1/2) <sup>∑</sup>*α*0=±<sup>1</sup> |*α*0��*α*0| ( and so �*α*1� = 0), then for all even *<sup>k</sup>*,

*<sup>k</sup>*−1*αk*�, �*αk*−1*αk*� and �*α*�

*<sup>k</sup>*� + �*α*�

measurements. So by using equations (43) and (13) we find that


Interestingly if *a*ˆ0 ⊥ *a*ˆ1 so that cos*θ*<sup>1</sup> = 0 ( and so, �*α*1�*QM* = 0) or, if the initial state is the

�*αn*−*<sup>k</sup>* ··· *<sup>α</sup>n*�*QM* = <sup>0</sup>

�*α*1*α*<sup>2</sup> ··· *αn*=2*p*+1�*QM* = 0. We shall now show that for *n* successive experiments ( with *n* > 1), QM violates the inequality |�*MKI*�| ≤ *<sup>s</sup>*<sup>3</sup> ( see equation (9)) up to <sup>√</sup>2 for *<sup>s</sup>* <sup>=</sup> 1/2 systems. We take the eigenvalues to be *α<sup>k</sup>* = ±1 so |�*Mk*�|*HVT* ≤ 1). We have already shown that for *n* = 2 and *n* = 3, the corresponding MKI's are violated (section 4). Now, we know that temporal two-fold

*<sup>k</sup>*−1*α*�

*<sup>k</sup>*−1*αk*�] + �*M*�

�

�

*<sup>k</sup>*� , �*α*�

Further,

and

and so

correlations �*αk*−1*α*�


2

∑ *α*0

⎧ ⎨ ⎩

$$\frac{\partial |\langle M\_k \rangle|}{\partial \phi\_{k-1}} = \frac{\partial |\langle M\_k \rangle|}{\partial \phi\_k} = \frac{\partial |\langle M\_k \rangle|}{\partial \phi\_{k-1}'} = \frac{\partial |\langle M\_k \rangle|}{\partial \phi\_k'} = 0 \Rightarrow \phi\_{k-1} = \phi\_{k-1}' = \phi\_k = \phi\_k'.$$

The maximum value of |�*Mk*�| will occur when all the vectors *<sup>a</sup>*ˆ*k*−1, *<sup>a</sup>*ˆ� *<sup>k</sup>*−1, *<sup>a</sup>*ˆ*k*, *<sup>a</sup>*ˆ� *<sup>k</sup>* lie on the same plane. We obtain:

$$\begin{split} |\langle M\_{k}\rangle| &\leq \frac{1}{2} |\langle M\_{k-2}\rangle| \cos(\theta\_{k-1} - \theta\_{k}^{\prime}) + \cos(\theta\_{k-1}^{\prime} - \theta\_{k}) | \\ &\quad + \langle M\_{k-2}^{\prime}\rangle [\cos(\theta\_{k-1} - \theta\_{k}) - (\cos\theta\_{k-1}^{\prime} - \theta\_{k}^{\prime})] | \\ &\leq \frac{1}{2} |\langle M\_{k-2}\rangle| [\cos(\theta\_{k-1} - \theta\_{k}^{\prime}) + \cos(\theta\_{k-1}^{\prime} - \theta\_{k})] \\ &\quad + \frac{1}{2} |\langle M\_{k-2}^{\prime}\rangle| [\cos(\theta\_{k-1} - \theta\_{k}) - (\cos\theta\_{k-1}^{\prime} - \theta\_{k}^{\prime})] | .\end{split} \tag{48}$$

By substituting *<sup>x</sup>* = *<sup>θ</sup>k*−<sup>1</sup> − *<sup>θ</sup>*� *<sup>k</sup>*, *y* = *θ*� *<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>k*, *<sup>z</sup>* <sup>=</sup> *<sup>θ</sup>k*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup><sup>k</sup>* and *<sup>θ</sup>*� *<sup>k</sup>*−<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*� *<sup>k</sup>* = *x* + *y* − *z* in above-equation and by using

$$\frac{\partial |\langle M\_k \rangle|}{\partial x} = \frac{\partial |\langle M\_k \rangle|}{\partial y} = \frac{\partial |\langle M\_k \rangle|}{\partial z} = 0,$$

we get *x* = *y* = −*z* = *π*/4. Finally, by using the fact |�*Mk*�| + |�*M*� *<sup>k</sup>*�| ≤ 2, we obtain:

$$|\langle M\_k \rangle| \le \frac{\sqrt{2}}{2} \{ |\langle M\_{k-2} \rangle + \langle M\_{k-2}' \rangle| \} \le \sqrt{2}.\tag{49}$$

One can get this result by induction hypothesis. Thus, we conclude that QM violates the MKI (|�*Mn*�| ≤ 1) for *<sup>n</sup>* successive measurements upto <sup>√</sup>2, *i.e.*,

$$
\eta\_{\rm n} = \sqrt{2}.\tag{50}
$$

#### **5.2 Violation Scarani-Gisin inequality (SCI)**

Although in contrast to correlations in space there are no genuine multi-mode correlations in time, we will see that temporal correlations can be stronger than spatial ones in a certain sense. We denote by max[*Bspace QM* (*i*, *j*)] the maximal value of the Bell expression for qubits *i* and *j* (Bell inequality is obtained by Eq(8)). Scarani and Gisin (Scarani & Gisin, 2001) found an interesting bound that holds for arbitrary state of three qubits:

$$\max[\mathbf{B}\_{\mathbf{QM}}^{\text{space}}(1,2)] + \max[\mathbf{B}\_{\mathbf{QM}}^{\text{space}}(2,3)] \le 2. \tag{51}$$

Spin Measurements 19

Quantum Correlations in Successive Spin Measurements 215

*b*2, ˆ

= 0 ⇒ *φ*<sup>1</sup> = *φ*<sup>2</sup> = ... = *φn*.

= <sup>0</sup> ⇒ *<sup>θ</sup>*<sup>12</sup> = *<sup>θ</sup>*<sup>23</sup> = ... = *<sup>θ</sup>*2*n*−1,2*<sup>n</sup>* = *<sup>θ</sup>*.

2n <sup>−</sup> cos (2n <sup>−</sup> <sup>1</sup>)<sup>ß</sup>

It would be interesting to consider Bell inequalities involving both two and three successive measurement correlations. The simplest way of obtaining such an inequality would be by adding genuinely bipartite correlations to the tripartite correlations considered in Mermin's inequality. For instance,a straightforward calculation would allow us to prove that any local

> 2*α*� 3�−�*α*�

> 2*α*� 3�−�*α*�

A numerical calculation shows that both the GHZ and W states give a same maximal violation of the inequality (59). However, if we assign a higher weight to the bipartite correlations

which is violated by the W state but not by GHZ state (Cabello, 2002b). It is not difficult to show that three successive measurements correlations for spin 1/2 break the hybrid Bell

> 2*α*� 3�−�*α*�

2*α*� 3�−�*α*�

So two successive measurements correlations are relevant to those of three successive

1*α*2*α*�

1*α*2*α*�

1*α*2*α*�

1*α*2*α*�

3�−�*α*� 1*α*� <sup>2</sup>*α*3�−

3�−�*α*� 1*α*� <sup>2</sup>*α*3�−

3�−�*α*� 1*α*� <sup>2</sup>*α*3�−

3�−�*α*� 1*α*� <sup>2</sup>*α*3�−

<sup>3</sup>�−�*α*2*α*3� ≤ 3 (59)

<sup>3</sup>� − 2�*α*2*α*3� ≤ 4, (60)

<sup>3</sup>�−�*α*2*α*3� ≤ 3.8 (61)

<sup>3</sup>� − 2�*α*2*α*3� ≤ 4.8. (62)

*b*4,..., ˆ

*CBI* = (2*n* − 1) cos *θ* − cos(2*n* − 1)*θ*. (56)

*∂θ* <sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>θ</sup>* <sup>=</sup> *<sup>π</sup>*/2*n*. (57)

2n <sup>=</sup> 2n cos <sup>ß</sup>

<sup>8</sup>*n*<sup>2</sup> ) for *n* −→ ∞. Therefore the maximum CBI can be made

*b*2*n*. The maximum value |*CBI*|

2n (58)

*<sup>a</sup>*ˆ1, *<sup>a</sup>*ˆ3,..., *<sup>a</sup>*ˆ2*<sup>n</sup>*−<sup>1</sup> and (*<sup>k</sup>* <sup>=</sup> 2, 4, . . . , 2*n*) for vectors <sup>ˆ</sup>

*∂*(CBI) *∂θ<sup>k</sup>*

CBI = (2n − 1) cos

<sup>2</sup>*<sup>n</sup>* ) tends to (<sup>1</sup> <sup>−</sup> *<sup>π</sup>*<sup>2</sup>

After partial differential over all *θk*, we get:

Therefore, we obtain:

We know cos( *<sup>π</sup>*

inequalities.

and

arbitrarily close to 2*n*.

By substituting , we obtain:

So,

will occur when all the vectors lie on the same plane. this is because of:

*∂*(CBI)

ß

**5.4 Violation Bell inequalities involving Tri and Bi-measurements correlations**

3�−�*α*1*α*�

3�−�*α*1*α*�

<sup>2</sup>� − 2�*α*1*α*�

3�−�*α*1*α*�

2�−�*α*1*α*�

3�−�*α*1*α*�

<sup>2</sup>� − 2�*α*1*α*�

measurements. This behavior is analogous to three particle W state (Cabello, 2002b).

2�−�*α*1*α*�

realistic theory must satisfy the following inequality (Cabello, 2002b):

�*α*1*α*�

appearing in the inequality, then we can reach a Bell inequality such as

2�*α*1*α*�

�*α*1*α*�

2�*α*1*α*�

−5 ≤ �*α*1*α*2*α*�

−8 ≤ �*α*1*α*2*α*�

−5.34 ≤ �*α*1*α*2*α*�

−8.2 ≤ �*α*1*α*2*α*�

*∂*(CBI) *∂φ<sup>k</sup>*

Physically, this means that no two pairs of qubits of a three-qubit system can violate the CHSH inequalities simultaneously. This is because if two systems are highly entangled, they can not be entangled highly to another systems. Let us denote by max[*Btime QM* (*i*, *j*)] the maximal value of the Bell expression for two consecutive observations of a single qubit at times *i* and *j*. Since quantum correlations between two successive measurements do not depend on the initial state ( see Eq (40)), one can obtain:

$$\begin{aligned} \left[B\_{QM}^{time}(k-1,k)\right] + \left[B\_{QM}^{time}(k,k+1)\right] &= \\ \frac{1}{2} \left[\cos\theta\_{k-1,k} + \cos\theta\_{k-1,k'} + \cos\theta\_{k-1',k} - \cos\theta\_{k-1',k'}\right] + \\ \frac{1}{2} \left[\cos\theta\_{k,k+1} + \cos\theta\_{k,k+1'} + \cos\theta\_{k',k+1} - \cos\theta\_{k',k+1'}\right]. \end{aligned} \tag{52}$$

By selecting,

$$
\theta\_{k-1,k} = \theta\_{k-1,k'} = \theta\_{k-1',k} = \theta\_{k,k+1} = \theta\_{k,k+1'} = \theta\_{k',k+1} = \frac{\pi}{4}
$$

and

$$
\theta\_{k-1',k'} = \theta\_{k'k+1'} = \frac{3\pi}{4},
$$

we obtain:

$$\max \left[ \mathbf{B}\_{\text{QM}}^{\text{time}} (\mathbf{k} - 1, \mathbf{k}) \right] + \max \left[ \mathbf{B}\_{\text{QM}}^{\text{time}} (\mathbf{k}, \mathbf{k} + 1) \right] = \sqrt{2} + \sqrt{2} = 2\sqrt{2} > 2. \tag{53}$$

Thus, although there are no genuine three-fold temporal correlations, a specific combination of two-fold correlations can have values that are not achievable with correlations in space for any three-qubit system. In fact, one would need two pairs of maximally entangled two-qubit states to achieve the bound in (53). Also note that the local realistic bound is 2, which is equal to the bound in (51). Similar conclusion can be obtained for the sum of *n* successive measurements.

$$\max\left[\mathbf{B}\_{\text{QM}}^{\text{time}}(1,2)\right] + \max\left[\mathbf{B}\_{\text{QM}}^{\text{time}}(2,3)\right] + \dots + \max\left[\mathbf{B}\_{\text{QM}}^{\text{time}}(\mathbf{n}-1,\mathbf{n})\right] = \mathbf{n}\sqrt{2} > \mathbf{n}.\tag{54}$$

#### **5.3 Violation chained Bell inequalities (CHI)**

Generalized CHSH inequalities may be obtained by providing more than two alternative experiments to each process. We consider two successive measurements on a spin- <sup>1</sup> <sup>2</sup> particle in a mixed state, such that the first experiment can measure spin component along one of the directions *<sup>a</sup>*ˆ1, *<sup>a</sup>*ˆ3,..., *<sup>a</sup>*ˆ2*<sup>n</sup>*−<sup>1</sup> and the second experiment along one of the directions <sup>ˆ</sup> *b*2, ˆ *b*4,..., ˆ *b*2*n*. The results of these measurements are called *α<sup>r</sup>* (*r* = 1, 3, . . . , 2*n* − 1) and *β<sup>s</sup>* (*s* = 2, 4, . . . , 2*n*), respectively, and their values are ±1 (in unit if ¯*h*/2). We have a generalized CHSH inequality (Braunstein & Caves, 1990),(Peres, 1993):

$$CBI = \frac{1}{2} \left| \langle a\_1 \beta\_2 \rangle + \langle \beta\_2 a\_3 \rangle + \langle a\_3 \beta\_4 \rangle + \dots + \langle a\_{2n-1} \beta\_{2n} \rangle - \langle \beta\_{2n} a\_1 \rangle \right| \le n - 1 \tag{55}$$

This upper bound is violated by quantum correlations in two successive measurements, increasingly with larger *n*. In order to obtain the maximum value above-inequality, we consider the spherical-polar co-ordinates (*θk*, *φk*),(*k* = 1, 3, . . . , 2*n* − 1) of the vectors *<sup>a</sup>*ˆ1, *<sup>a</sup>*ˆ3,..., *<sup>a</sup>*ˆ2*<sup>n</sup>*−<sup>1</sup> and (*<sup>k</sup>* <sup>=</sup> 2, 4, . . . , 2*n*) for vectors <sup>ˆ</sup> *b*2, ˆ *b*4,..., ˆ *b*2*n*. The maximum value |*CBI*| will occur when all the vectors lie on the same plane. this is because of:

$$\frac{\partial(\text{CBI})}{\partial \phi\_k} = 0 \Rightarrow \phi\_1 = \phi\_2 = \dots = \phi\_n.$$

After partial differential over all *θk*, we get:

$$\frac{\partial(\text{CBI})}{\partial \theta\_k} = 0 \Rightarrow \theta\_{12} = \theta\_{23} = \dots = \theta\_{2n-1, 2n} = \theta.$$

Therefore, we obtain:

$$
\mathbb{C}BI = (2n - 1)\cos\theta - \cos(2n - 1)\theta. \tag{56}
$$

So,

18 Will-be-set-by-IN-TECH

Physically, this means that no two pairs of qubits of a three-qubit system can violate the CHSH inequalities simultaneously. This is because if two systems are highly entangled, they can not

of the Bell expression for two consecutive observations of a single qubit at times *i* and *j*. Since quantum correlations between two successive measurements do not depend on the initial

*QM* (*k*, *k* + 1)

 =

,*<sup>k</sup>* = *θk*,*k*+<sup>1</sup> = *θk*,*k*+1� = *θk*�

,*k*+1� <sup>=</sup> <sup>3</sup>*<sup>π</sup>*

QM (k, k + 1)

+ ... + max

Generalized CHSH inequalities may be obtained by providing more than two alternative

in a mixed state, such that the first experiment can measure spin component along one of the directions *<sup>a</sup>*ˆ1, *<sup>a</sup>*ˆ3,..., *<sup>a</sup>*ˆ2*<sup>n</sup>*−<sup>1</sup> and the second experiment along one of the directions <sup>ˆ</sup>

(*s* = 2, 4, . . . , 2*n*), respectively, and their values are ±1 (in unit if ¯*h*/2). We have a generalized

This upper bound is violated by quantum correlations in two successive measurements, increasingly with larger *n*. In order to obtain the maximum value above-inequality, we consider the spherical-polar co-ordinates (*θk*, *φk*),(*k* = 1, 3, . . . , 2*n* − 1) of the vectors

*b*2*n*. The results of these measurements are called *α<sup>r</sup>* (*r* = 1, 3, . . . , 2*n* − 1) and *β<sup>s</sup>*

<sup>2</sup> |�*α*1*β*2� <sup>+</sup> �*β*2*α*3� <sup>+</sup> �*α*3*β*4� <sup>+</sup> ... <sup>+</sup> �*α*2*n*−1*β*2*n*�−�*β*2*nα*1�| <sup>≤</sup> *<sup>n</sup>* <sup>−</sup> <sup>1</sup>

Thus, although there are no genuine three-fold temporal correlations, a specific combination of two-fold correlations can have values that are not achievable with correlations in space for any three-qubit system. In fact, one would need two pairs of maximally entangled two-qubit states to achieve the bound in (53). Also note that the local realistic bound is 2, which is equal to the bound in (51). Similar conclusion can be obtained for the sum of *n* successive

4 ,

 <sup>=</sup> <sup>√</sup>

 Btime

,*<sup>k</sup>* − cos *<sup>θ</sup>k*−1�

<sup>2</sup> <sup>+</sup> <sup>√</sup>

QM (n − 1, n)

,*k*+<sup>1</sup> − cos *θk*�

,*k*� +

,*k*+<sup>1</sup> <sup>=</sup> *<sup>π</sup>* 4

> 2 = 2 √

> > = n √

,*k*+1� 

*QM* (*i*, *j*)] the maximal value

. (52)

2 > 2. (53)

2 > n. (54)

<sup>2</sup> particle

(55)

be entangled highly to another systems. Let us denote by max[*Btime*

*QM* (*<sup>k</sup>* <sup>−</sup> 1, *<sup>k</sup>*)] + [*Btime*

*<sup>θ</sup>k*−1,*<sup>k</sup>* = *<sup>θ</sup>k*−1,*k*� = *<sup>θ</sup>k*−1�

cos *<sup>θ</sup>k*−1,*<sup>k</sup>* + cos *<sup>θ</sup>k*−1,*k*� + cos *<sup>θ</sup>k*−1�

cos *θk*,*k*+<sup>1</sup> + cos *θk*,*k*+1� + cos *θk*�

*θk*−1�

QM (<sup>k</sup> <sup>−</sup> 1, k)] + max[Btime

,*k*� = *θk*�

experiments to each process. We consider two successive measurements on a spin- <sup>1</sup>

state ( see Eq (40)), one can obtain:

By selecting,

we obtain:

measurements.

max Btime QM (1, 2)

*b*4,..., ˆ

*CBI* <sup>=</sup> <sup>1</sup>

*b*2, ˆ

max Btime

and

 *Btime*

1 2 

1 2 

> + max Btime QM (2, 3)

**5.3 Violation chained Bell inequalities (CHI)**

CHSH inequality (Braunstein & Caves, 1990),(Peres, 1993):

$$\frac{\partial(\text{CBI})}{\partial\theta} = 0 \Rightarrow \theta = \pi/2\text{n.}\tag{57}$$

By substituting , we obtain:

$$\text{CBI} = (2\text{n} - 1)\cos\frac{\text{g}}{2\text{n}} - \cos\frac{(2\text{n} - 1)\text{g}}{2\text{n}} = 2\text{n}\cos\frac{\text{g}}{2\text{n}}\tag{58}$$

We know cos( *<sup>π</sup>* <sup>2</sup>*<sup>n</sup>* ) tends to (<sup>1</sup> <sup>−</sup> *<sup>π</sup>*<sup>2</sup> <sup>8</sup>*n*<sup>2</sup> ) for *n* −→ ∞. Therefore the maximum CBI can be made arbitrarily close to 2*n*.

#### **5.4 Violation Bell inequalities involving Tri and Bi-measurements correlations**

It would be interesting to consider Bell inequalities involving both two and three successive measurement correlations. The simplest way of obtaining such an inequality would be by adding genuinely bipartite correlations to the tripartite correlations considered in Mermin's inequality. For instance,a straightforward calculation would allow us to prove that any local realistic theory must satisfy the following inequality (Cabello, 2002b):

$$-5 \le \langle \alpha\_1 a\_2 a\_3' \rangle - \langle \alpha\_1 a\_2' a\_3' \rangle - \langle a\_1' a\_2 a\_3' \rangle - \langle a\_1' a\_2' a\_3 \rangle -$$

$$\langle a\_1 a\_2' \rangle - \langle a\_1 a\_3' \rangle - \langle a\_2 a\_3 \rangle \le 3 \tag{59}$$

A numerical calculation shows that both the GHZ and W states give a same maximal violation of the inequality (59). However, if we assign a higher weight to the bipartite correlations appearing in the inequality, then we can reach a Bell inequality such as

$$-8 \le \langle \alpha\_1 a\_2 a\_3' \rangle - \langle \alpha\_1 a\_2' a\_3' \rangle - \langle a\_1' a\_2 a\_3' \rangle - \langle a\_1' a\_2' a\_3 \rangle -$$

$$2 \langle a\_1 a\_2' \rangle - 2 \langle a\_1 a\_3' \rangle - 2 \langle a\_2 a\_3 \rangle \le 4,\tag{60}$$

which is violated by the W state but not by GHZ state (Cabello, 2002b). It is not difficult to show that three successive measurements correlations for spin 1/2 break the hybrid Bell inequalities.

$$-5.34 \le \langle a\_1 a\_2 a\_3' \rangle - \langle a\_1 a\_2' a\_3' \rangle - \langle a\_1' a\_2 a\_3' \rangle - \langle a\_1' a\_2' a\_3 \rangle -$$

$$\langle a\_1 a\_2' \rangle - \langle a\_1 a\_3' \rangle - \langle a\_2 a\_3 \rangle \le 3.8 \tag{61}$$

and

$$-8.2 \le \langle a\_1 a\_2 a\_3' \rangle - \langle a\_1 a\_2' a\_3' \rangle - \langle a\_1' a\_2 a\_3' \rangle - \langle a\_1' a\_2' a\_3 \rangle -$$

$$2 \langle a\_1 a\_2' \rangle - 2 \langle a\_1 a\_3' \rangle - 2 \langle a\_2 a\_3 \rangle \le 4.8. \tag{62}$$

So two successive measurements correlations are relevant to those of three successive measurements. This behavior is analogous to three particle W state (Cabello, 2002b).

Spin Measurements 21

Quantum Correlations in Successive Spin Measurements 217

�

<sup>2</sup> = *j*,...,*s*.*a*ˆ*<sup>n</sup>* = *j* − 1) = 0,

<sup>2</sup> = *j*,...,*s*.*a*ˆ*<sup>n</sup>* = *j* − 2) = 0,

<sup>2</sup> = *j*,...,*s*.*a*ˆ*<sup>n</sup>* = −*j*) = 0,

�

<sup>2</sup> = *j*,...,*s*.*a*ˆ

First, we prove here that all time-local SHVTs predict *p* = 0. Suppose that a time-local SHVT reproducing, in accordance with Eq.(4), the quantum predictions exist. Accordingly, if we

where the second equality is implied by the time-locality condition of Eq.(4). The last equality

*a* � *<sup>n</sup>* = *j*)

<sup>1</sup> <sup>=</sup> *<sup>j</sup>*,...*s*.*a*ˆ*<sup>l</sup>* <sup>=</sup> *<sup>j</sup>* <sup>−</sup> 1, . . . ,*s*. <sup>ˆ</sup>

<sup>1</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*.*a*ˆ*<sup>l</sup>* <sup>=</sup> *<sup>j</sup>* <sup>−</sup> <sup>1</sup>)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

= 0, (69)

*a* �

*p<sup>λ</sup>* (*s*.*a*ˆ1 = *j*)*pλ*(*s*.*a*ˆ2 = *j*)... *pλ*(*s*.*a*ˆ*<sup>n</sup>* = *j*) = 0, (70)

*a* �

> *a* �

> *a* �

*a* � *<sup>n</sup>* = *j*) = 0,

*<sup>n</sup>* = *j*) = 0,

*a* �

*a* �

<sup>2</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

<sup>2</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

<sup>1</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*.*a*ˆ*<sup>l</sup>* <sup>=</sup> *<sup>j</sup>* <sup>−</sup> <sup>1</sup>)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

<sup>1</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*.*a*ˆ*<sup>l</sup>* <sup>=</sup> *<sup>j</sup>* <sup>−</sup> <sup>2</sup>)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

<sup>1</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*.*a*ˆ*<sup>l</sup>* <sup>=</sup> <sup>−</sup>*j*)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

*<sup>n</sup>* = *j*) = 0, (67)

*<sup>n</sup>* = *j*) = *p*. (68)

*a* � *<sup>n</sup>* = *j*)

<sup>1</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

*<sup>n</sup>* = *j*) = 0,

*<sup>n</sup>* = *j*) = 0,

*a* �

*<sup>n</sup>* = *j*) = 0, (71)

*<sup>n</sup>* = *j*) = 0, (72)

*<sup>n</sup>* = *j*) vavishes

*a* � *<sup>n</sup>* = *j*)

<sup>1</sup> = *j*,...*s*.*a*ˆ*<sup>l</sup>* = −*j*,...,*s*.*a*ˆ

. . *p* (*s*.*a*ˆ �

. . . *p* (*s*.*a*ˆ � <sup>1</sup> = *j*,*s*.*a*ˆ �

*p* (*s*.*a*ˆ � <sup>1</sup> = *j*,*s*.*a*ˆ �

. . *p* (*s*.*a*ˆ � <sup>1</sup> = *j*,*s*.*a*ˆ �

*p* (*s*.*a*ˆ � <sup>1</sup> = *j*,*s*.*a*ˆ �

*<sup>d</sup>λρ*(*λ*)*pλ*(*s*. <sup>ˆ</sup>

*<sup>d</sup>λρ*(*λ*)*pλ*(*s*. <sup>ˆ</sup>

in Eq.(69) can be fulfilled if and only if the product *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

*<sup>p</sup><sup>λ</sup>* (*s*.*a*ˆ1 <sup>=</sup> *<sup>j</sup>* <sup>−</sup> <sup>1</sup>)*pλ*(*s*. <sup>ˆ</sup>

*<sup>p</sup><sup>λ</sup>* (*s*.*a*ˆ1 <sup>=</sup> *<sup>j</sup>* <sup>−</sup> <sup>2</sup>)*pλ*(*s*. <sup>ˆ</sup>

*<sup>p</sup><sup>λ</sup>* (*s*.*a*ˆ1 <sup>=</sup> <sup>−</sup>*j*)*pλ*(*s*. <sup>ˆ</sup>

. .

. . . *<sup>p</sup><sup>λ</sup>* (*s*. <sup>ˆ</sup> *a* �

*<sup>p</sup><sup>λ</sup>* (*s*. <sup>ˆ</sup> *a* �

. . *<sup>p</sup><sup>λ</sup>* (*s*. <sup>ˆ</sup> *a* �

<sup>1</sup> <sup>=</sup> *<sup>j</sup>*,...,*s*.*a*ˆ*<sup>l</sup>* <sup>=</sup> *<sup>j</sup>* <sup>−</sup> 1, . . . ,*s*. <sup>ˆ</sup>

*a* �

*a* �

every time within Λ. An equivalent result holds for Eqs.(64-69), leading to:

*a* �

*a* �

*a* �

consider, for example, Eq.(64), we must have

*p*(*s*. ˆ *a* �

> = Λ

> = Λ

#### **6. Hardy's argument for** *n* **successive measurements for all spin-***s* **measurements**

Hardy's nonlocality argument is considered weaker than Bell inequalities in the bipartite case, as every maximally entangled state of two spin- <sup>1</sup> <sup>2</sup> particles violates Bell's inequality maximally but none of them satisfies Hardy-type nonlocality conditions. The scenario in successive spin measurements is quite different, however. We showed in previous sections that all *n* successive spin-*s* measurements break Bell-type inequalities, in contrast to the bipartite case, where only the entangled states break it. In this section, we prove that all *n* successive spin-*s* measurements satisfy Hardy-type argument conditions. Consider four yes/no-type events *A*, *A*� , *B* and *B* � , where *A* and *A*� may happen at time *t*1, and *B* and *B* � may happen at another time, *t*<sup>2</sup> (*t*<sup>2</sup> > *t*1). The joint probability that, at the first time (*t*1), *A* and, at the second time (*t*2), B are " yes" is 0. The joint probability that, at the first time (*t*1), *A* is "no" and, at the second time (*t*2), *B* � is "yes" is 0. The joint probability that, at the first time (*t*1), *A*� is "yes" and, at the second time (*t*2), *B* is "no", is 0. The joint probability that both *A*� and *B* � are "yes" is nonzero. We can write this as follows:

$$p(A=+1, B=+1) = 0,$$

$$\begin{split}p(A=-1, B'=+1) &= 0, \\ p(A'=+1, B=-1) &= 0, \\ p(A'=+1, \dot{B}=+1) &= p \neq 0. \end{split} \tag{63}$$

We show that these four statements are not compatible with time-local realism. The nonzero probability appearing in the argument is the measure of violation of time-local realism. It is interesting that two successive s-spin measurements violate time-local realism. We deal with the case where the input state is a pure state whose eigenstates coincide with those of *s*.*a*ˆ0 for some *a*ˆ0 whose eigenvalues we denote *α*<sup>0</sup> = *j*. Hardy's argument for a system of *n* successive spin-*s* measurements, in it's minimal form (Parasuram & Ghosh, n.d.), is given by following conditions:

$$\begin{aligned} p(s.\hat{\mathbf{a}}\_1 = j, \mathbf{s}.\hat{\mathbf{a}}\_2 = j, \dots, s.\hat{\mathbf{a}}\_n = j) &= 0, & \quad (64) \\ p(s.\hat{\mathbf{a}}\_1 = j - 1, s.\hat{\mathbf{a}}\_2' = j, \dots, s.\hat{\mathbf{a}}\_n' = j) &= 0, \\ p(s.\hat{\mathbf{a}}\_1 = j - 2, s.\hat{\mathbf{a}}\_2' = j, \dots, s.\hat{\mathbf{a}}\_n' = j) &= 0, \\ \\ \vdots \\ p(s.\hat{\mathbf{a}}\_1 = -j, \mathbf{s}.\hat{\mathbf{a}}\_2' = j, \dots, \mathbf{s}.\hat{\mathbf{a}}\_n' = j) &= 0, \\ \\ \vdots \\ p(s.\hat{\mathbf{a}}\_1' = j, \dots, \mathbf{s}.\hat{\mathbf{a}}\_l = j - 1, \dots, \mathbf{s}.\hat{\mathbf{a}}\_n' = j) &= 0, \\ p(s.\hat{\mathbf{a}}\_1' = j, \dots, \mathbf{s}.\hat{\mathbf{a}}\_l = j - 2, \dots, \mathbf{s}.\hat{\mathbf{a}}\_n' = j) &= 0, \\ \end{aligned} \tag{65}$$

.

20 Will-be-set-by-IN-TECH

**6. Hardy's argument for** *n* **successive measurements for all spin-***s* **measurements** Hardy's nonlocality argument is considered weaker than Bell inequalities in the bipartite case,

but none of them satisfies Hardy-type nonlocality conditions. The scenario in successive spin measurements is quite different, however. We showed in previous sections that all *n* successive spin-*s* measurements break Bell-type inequalities, in contrast to the bipartite case, where only the entangled states break it. In this section, we prove that all *n* successive spin-*s* measurements satisfy Hardy-type argument conditions. Consider four yes/no-type events

time, *t*<sup>2</sup> (*t*<sup>2</sup> > *t*1). The joint probability that, at the first time (*t*1), *A* and, at the second time (*t*2), B are " yes" is 0. The joint probability that, at the first time (*t*1), *A* is "no" and, at the second

is "yes" is 0. The joint probability that, at the first time (*t*1), *A*�

*p*(*A* = +1, *B* = +1) = 0,

�

= +1, *B* = −1) = 0,

We show that these four statements are not compatible with time-local realism. The nonzero probability appearing in the argument is the measure of violation of time-local realism. It is interesting that two successive s-spin measurements violate time-local realism. We deal with the case where the input state is a pure state whose eigenstates coincide with those of *s*.*a*ˆ0 for some *a*ˆ0 whose eigenvalues we denote *α*<sup>0</sup> = *j*. Hardy's argument for a system of *n* successive spin-*s* measurements, in it's minimal form (Parasuram & Ghosh, n.d.), is given by following

*p*(*A* = −1, *B*

*p*(*A*�

*p*(*A*�

*p*(*s*.*a*ˆ1 = *j* − 1,*s*.*a*ˆ

*p*(*s*.*a*ˆ1 = *j* − 2,*s*.*a*ˆ

*p*(*s*.*a*ˆ1 = −*j*,*s*.*a*ˆ

. .

. .

*p*(*s*.*a*ˆ �

*p*(*s*.*a*ˆ � �

�

�

<sup>2</sup> = *j*,...,*s*.*a*ˆ

<sup>2</sup> = *j*,...,*s*.*a*ˆ

<sup>2</sup> = *j*,...,*s*.*a*ˆ

<sup>1</sup> = *j*,...*s*.*a*ˆ*<sup>l</sup>* = *j* − 1, . . . ,*s*.*a*ˆ

<sup>1</sup> = *j*,...,*s*.*a*ˆ*<sup>l</sup>* = *j* − 2, . . . ,*s*.*a*ˆ

second time (*t*2), *B* is "no", is 0. The joint probability that both *A*�

may happen at time *t*1, and *B* and *B*

= +1) = 0,

<sup>2</sup> particles violates Bell's inequality maximally

�

and *B* �

= +1, *<sup>B</sup>*´ = +1) = *<sup>p</sup>* �<sup>=</sup> 0. (63)

*p*(*s*.*a*ˆ1 = *j*,*s*.*a*ˆ2 = *j*,...,*s*.*a*ˆ*<sup>n</sup>* = *j*) = 0, (64)

*<sup>n</sup>* = *j*) = 0,

*<sup>n</sup>* = *j*) = 0,

*<sup>n</sup>* = *j*) = 0,

. (65)

*<sup>n</sup>* = *j*) = 0,

*<sup>n</sup>* = *j*) = 0,

�

�

�

�

�

may happen at another

is "yes" and, at the

(66)

are "yes" is nonzero.

as every maximally entangled state of two spin- <sup>1</sup>

, where *A* and *A*�

*A*, *A*�

time (*t*2), *B*

conditions:

, *B* and *B* �

�

We can write this as follows:

$$\begin{cases} p \text{ (s.a}\_1' = j, \dots s.\text{a.}\_\text{l} = -j, \dots, s.\text{a.}\_\text{l}' = j) = 0, \\\\ \text{ } & \text{ } \\\\ p \text{ (s.a}\_1' = j, \text{s.a}\_2' = j, \dots, \text{s.a.}\_\text{l} = j - 1) = 0, \\\\ p \text{ (s.a}\_1' = j, \text{s.a}\_2' = j, \dots, \text{s.a.}\_\text{l} = j - 2) = 0, \\\\ \text{ } & \text{ } \\\\ p \text{ (s.a}\_1' = j, \text{s.a.}\_2' = j, \dots, \text{s.a.}\_\text{l} = -j) = 0, \\\\ p \text{ (s.a}\_1' = j, \text{s.a.}\_2' = j, \dots, \text{s.a.}\_\text{l} = j) = p. \end{cases}$$

First, we prove here that all time-local SHVTs predict *p* = 0. Suppose that a time-local SHVT reproducing, in accordance with Eq.(4), the quantum predictions exist. Accordingly, if we consider, for example, Eq.(64), we must have

$$\begin{aligned} p(\text{s.a}^{\uparrow}\_{1} = j, \dots, \text{s.a}\_{l} = j - 1, \dots, \text{s.a}^{\uparrow}\_{n} = j) \\ = \int\_{\Lambda} d\lambda \rho(\lambda) p\_{\lambda}(\text{s.a}^{\uparrow}\_{1} = j, \dots \text{s.a}\_{l} = j - 1, \dots, \text{s.a}^{\uparrow}\_{n} = j) \\ = \int\_{\Lambda} d\lambda \rho(\lambda) p\_{\lambda}(\text{s.a}^{\uparrow}\_{1} = j) \dots p\_{\lambda}(\text{s.a}\_{l} = j - 1) \dots p\_{\lambda}(\text{s.a}^{\uparrow}\_{n} = j) \\ = 0, \end{aligned} \tag{69}$$

where the second equality is implied by the time-locality condition of Eq.(4). The last equality in Eq.(69) can be fulfilled if and only if the product *<sup>p</sup>λ*(*s*. <sup>ˆ</sup> *a* � <sup>1</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup> *a* � *<sup>n</sup>* = *j*) vavishes every time within Λ. An equivalent result holds for Eqs.(64-69), leading to:

$$\begin{aligned} \label{eq:1} \rho\_{\lambda}(\text{s.\hat{a}}\_{1}=j) p\_{\lambda}(\text{s.\hat{a}}\_{2}=j) & \dots p\_{\lambda}(\text{s.\hat{a}}\_{n}=j) = 0, \\ p\_{\lambda}(\text{s.\hat{a}}\_{1}=j-1) p\_{\lambda}(\text{s.\hat{a}}\_{2}=j) & \dots p\_{\lambda}(\text{s.\hat{a}}\_{n}=j) = 0, \\ p\_{\lambda}(\text{s.\hat{a}}\_{1}=j-2) p\_{\lambda}(\text{s.\hat{a}}\_{2}=j) & \dots p\_{\lambda}(\text{s.\hat{a}}\_{n}=j) = 0, \\\\ \begin{array}{c} \\ \text{.} \\ p\_{\lambda}\left(\text{s.\hat{a}}\_{1}=-j\right) p\_{\lambda}(\text{s.\hat{a}}\_{2}=j) & \dots p\_{\lambda}(\text{s.\hat{a}}\_{n}=j) = 0, \\ \end{array} \\\\ \begin{array}{c} \\ \text{.} \\ p\_{\lambda}\left(\text{s.\hat{a}}\_{1}=j\right) & \dots p\_{\lambda}(\text{s.\hat{a}}\_{l}=j-1) \dots p\_{\lambda}(\text{s.\hat{a}}\_{n}=j) = 0, \\ p\_{\lambda}\left(\text{s.\hat{a}}\_{1}=j\right) & \dots p\_{\lambda}(\text{s.\hat{a}}\_{l}=j-2) \dots p\_{\lambda}(\text{s.\hat{a}}\_{n}=j) = 0, \\ \end{array} \end{aligned} \tag{71}$$

Spin Measurements 23

Quantum Correlations in Successive Spin Measurements 219

belongs to *Al*, then, by definition, *pλ*(*s*.*a*ˆ*<sup>l</sup>* = *j*) = 0, so that Eq.(75) can be satisfied only if

a result leading to a contradiction of Eq.(74), which requires that there is a set of nonzero *ρ* measure within Λ where both probabilities do not vanish. To summarize, we have shown that it is not possible to exhibit any time-local hidden-variable model, satisfying Hardy's logic

Now, we show that in quantum theory for the *n* successive spin measurement, sometimes *p* > 0. So, we consider *n* successive measurements in directions *s*.*a*ˆ*<sup>i</sup>* (*i* = 1, 2, . . . , *n*) on spin-*s*

where *β<sup>k</sup>* is the angle between the *a*ˆ*<sup>k</sup>* and the +*z* axes. So, given the input state |*α*0�, the ( joint) probability that the measurement outcomes will be |*α*1�∈{+*j*,..., −*j*} in the first measurement, |*α*2�∈{+*j*,..., −*j*} in the second measurement,..., |*αn*�∈{+*j*,..., −*j*} in the

> =Π*<sup>n</sup> k*=1*d*<sup>2</sup> *αk*−1,*α<sup>k</sup>*

We deal with the case where the input state is a pure state whose eigenstates coincide with

<sup>2</sup> <sup>−</sup> *<sup>β</sup>*1)... *<sup>d</sup>*<sup>2</sup>

<sup>2</sup> <sup>−</sup> *<sup>β</sup>*1)... *<sup>d</sup>*<sup>2</sup>

<sup>2</sup> <sup>−</sup> *<sup>β</sup>*1)... *<sup>d</sup>*<sup>2</sup>

� *<sup>l</sup>*−1)*d*<sup>2</sup>

� *<sup>l</sup>*−1)*d*<sup>2</sup>

*<sup>j</sup>*,*j*−1(*β<sup>l</sup>* <sup>−</sup> *<sup>β</sup>*

*<sup>j</sup>*,*j*−2(*β<sup>l</sup>* <sup>−</sup> *<sup>β</sup>*

(*β<sup>l</sup>* − *β* � *<sup>l</sup>*−1)*d*<sup>2</sup> *j*,−*j* (*β* �

*<sup>n</sup>* = *j*) = 0. Hence, for any *λ* ∈ {*A*<sup>1</sup> ∪ *A*<sup>2</sup> ∪ ... ∪ *An*}, we obtain

(*s*)

*<sup>k</sup>*=1|�*αk*−1|*αk*�|

*S*.*a*ˆ0 for some *a*ˆ0 whose eigenvalues we denote *α*<sup>0</sup> = *j*. Now, by substituting Eq.(79)

*jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0,

*jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0,

*jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0,

*<sup>j</sup>*,*j*−1(*<sup>β</sup>* �

*<sup>j</sup>*,*j*−2(*<sup>β</sup>* �

*<sup>l</sup>*+<sup>1</sup> − *βl*)...

*<sup>l</sup>*+<sup>1</sup> − *βl*)...

*<sup>l</sup>*+<sup>1</sup> − *βl*)...

*<sup>n</sup>*−1) = 0, (81)

*<sup>α</sup>k*−1,*α<sup>k</sup>* (*β<sup>k</sup>* − *<sup>β</sup>k*−1), (78)

(*β<sup>k</sup>* − *<sup>β</sup>k*−1). (79)

*jj*(*β<sup>n</sup>* − *<sup>β</sup>n*−1) = 0, (80)

*<sup>p</sup>λ*(*s*. <sup>ˆ</sup> *a* �

those of �

<sup>1</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup>

for *n* successive measurements.

*n*-th measurement, is given by

*a* �

particles. For a spin-*s* system, we have ( see Appendix-B):


*p*(*α*1, *α*2,..., *αn*) =Π*<sup>n</sup>*

in the minimal form of Hardy's argument [ Eqs.(64)-(68)], we have

*<sup>j</sup>*,*j*−1(*β*1)*d*<sup>2</sup>

*<sup>j</sup>*,*j*−2(*β*1)*d*<sup>2</sup>

(*β*1)*d*<sup>2</sup> *j*,−*j* (*β* �

*jj*(*β*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*1)... *<sup>d</sup>*<sup>2</sup>

*<sup>j</sup>*,*j*−1(*<sup>β</sup>* �

*<sup>j</sup>*,*j*−2(*<sup>β</sup>* �

*d* 2 *jj*(*β*1)*d*<sup>2</sup>

*d* 2

*d* 2

. . *d* 2 *j*,−*j*

. . *d* 2 *jj*(*β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup>

*d* 2 *jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0,

*d* 2 *jj*(*β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup>

*d* 2 *jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0,

*d* 2 *jj*(*β* � *<sup>n</sup>* − *β* �

. . *d* 2 *jj*(*β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup> *j*,−*j*

$$\begin{aligned} p\_{\lambda} \ (\text{s.a}\_{1}^{\prime} = j) p\_{\lambda} (\text{s.a}\_{2}^{\prime} = j) \dots p\_{\lambda} (\text{s.a}\_{n} = j - 1) &= 0, \\ p\_{\lambda} \ (\text{s.a}\_{1}^{\prime} = j) p\_{\lambda} (\text{s.a}\_{2}^{\prime} = j) \dots p\_{\lambda} (\text{s.a}\_{n} = j - 2) &= 0, \\ \ . \ . \\ p\_{\lambda} \ (\text{s.a}\_{1}^{\prime} = j) p\_{\lambda} (\text{s.a}\_{2}^{\prime} = j) \dots p\_{\lambda} (\text{s.a}\_{n} = -j) &= 0, \end{aligned}$$

$$p\_{\lambda} \quad (\text{s.d.} = \mathfrak{j})\\p\_{\lambda}(\text{s.d.}\_2 = \mathfrak{j}) \dots p\_{\lambda}(\text{s.d.}\_n = -\mathfrak{j}) = \mathfrak{0},\tag{73}$$

$$p\_{\lambda} \text{ (s.d.}\_{1}^{\circ} = j \text{)} \\ p\_{\lambda} (\text{s.d.}\_{2}^{\circ} = j) \dots p\_{\lambda} (\text{s.d.}\_{n}^{\circ} = j) = p \neq 0,\tag{74}$$

where the first 2*jn* + 1 equations are supposed to hold almost every time within Λ, while the last equation has to be satisfied in a subset of Λ whose measure according to the distribution *ρ*(*λ*) is nonzero. To prove the more general result that no conceivable time-local SHVT can simultaneously satisfy Eqs.(70)-(74), a manipulation of those equations is required. To this end, let us sum all equations in each set. We obtain

$$\begin{aligned} \left(1 - p\_{\lambda}(s.\hat{\mathfrak{a}}\_{1} = j)\right) \left[p\_{\lambda}(s.\hat{\mathfrak{a}}\_{2} = j) \dots p\_{\lambda}(s.\hat{\mathfrak{a}}\_{n}^{\prime} = j)\right] &= 0, \\\\ \left. \cdot \right. \\\\ \left(1 - p\_{\lambda}(s.\hat{\mathfrak{a}}\_{l} = j)\right) \left[p\_{\lambda}(s.\hat{\mathfrak{a}}\_{1}^{\prime} = j) \dots p\_{\lambda}(s.\hat{\mathfrak{a}}\_{n}^{\prime} = j)\right] &= 0, \\\\ \left. \cdot \right. \\\\ \left(1 - p\_{\lambda}(s.\hat{\mathfrak{a}}\_{n} = j)\right) \left[p\_{\lambda}(s.\hat{\mathfrak{a}}\_{1}^{\prime} = j) \dots p\_{\lambda}(s.\hat{\mathfrak{a}}\_{n-1}^{\prime} = j)\right] &= 0. \end{aligned} \tag{75}$$

Now let us partition the set of hidden variables Λ and define the following subsets *A*1, *A*2,... *An*, and *B* as:

$$\begin{aligned} A\_1 &= \{ \lambda \in \Lambda | p\_\lambda(\text{s.a.}\_1 = j) = 0 \}, \\\\ \cdot &\quad . \\ A\_l &= \{ \lambda \in \Lambda | p\_\lambda(\text{s.a.}\_l = j) = 0 \}, \\\\ \cdot &\quad . \\\\ A\_{ll} &= \{ \lambda \in \Lambda | p\_\lambda(\text{s.a.}\_n = j) = 0 \}, \\ B &= \Lambda - \{ A\_1 \cup A\_2 \cup \dots \cup A\_n \}. \end{aligned} \tag{77}$$

We have that, for all *λ* belonging to B, *pλ*(*s*.*a*ˆ1 = *j*)*pλ*(*s*.*a*ˆ2 = *j*)... *pλ*(*s*.*a*ˆ*<sup>n</sup>* = *j*) �= 0.If set B had a nonzero measure according to the distribution *ρ*, that is, if *<sup>B</sup> dλρ*(*λ*) �= 0, there would be violation of Eq.(70) and, consequently, of Eq.(64). Therefore, to fuifill Eq.(70), the set *A*<sup>1</sup> ∪ *A*<sup>2</sup> ∪ ... ∪ *An* must coincide with Λ apart from a set of zero measure, and we are left only with hidden variables belonging to either *A*<sup>1</sup> or *A*<sup>2</sup> or . . . or *An*. If *λ*

. . . 22 Will-be-set-by-IN-TECH

<sup>2</sup> = *j*)... *pλ*(*s*.*a*ˆ*<sup>n</sup>* = *j* − 1) = 0,

<sup>2</sup> = *j*)... *pλ*(*s*.*a*ˆ*<sup>n</sup>* = *j* − 2) = 0,

�

<sup>2</sup> = *j*)... *pλ*(*s*.*a*ˆ

<sup>1</sup> = *j*)... *pλ*(*s*.*a*ˆ

<sup>1</sup> = *j*)... *pλ*(*s*.*a*ˆ

Now let us partition the set of hidden variables Λ and define the following subsets

*A*<sup>1</sup> = {*λ* ∈ Λ|*pλ*(*s*.*a*ˆ1 = *j*) = 0},

*An* = {*λ* ∈ Λ|*pλ*(*s*.*a*ˆ*<sup>n</sup>* = *j*) = 0},

We have that, for all *λ* belonging to B, *pλ*(*s*.*a*ˆ1 = *j*)*pλ*(*s*.*a*ˆ2 = *j*)... *pλ*(*s*.*a*ˆ*<sup>n</sup>* = *j*) �= 0.If set

would be violation of Eq.(70) and, consequently, of Eq.(64). Therefore, to fuifill Eq.(70), the set *A*<sup>1</sup> ∪ *A*<sup>2</sup> ∪ ... ∪ *An* must coincide with Λ apart from a set of zero measure, and we are left only with hidden variables belonging to either *A*<sup>1</sup> or *A*<sup>2</sup> or . . . or *An*. If *λ*

� *<sup>n</sup>* = *j*) = 0,

� *<sup>n</sup>* = *j*) = 0,

� *<sup>n</sup>*−<sup>1</sup> <sup>=</sup> *<sup>j</sup>*)

*Al* = {*λ* ∈ Λ|*pλ*(*s*.*a*ˆ*<sup>l</sup>* = *j*) = 0}, (76)

*B* = Λ − {*A*<sup>1</sup> ∪ *A*<sup>2</sup> ∪ ... ∪ *An*}. (77)

= 0. (75)

*<sup>B</sup> dλρ*(*λ*) �= 0, there

<sup>2</sup> = *j*)... *pλ*(*s*.*a*ˆ*<sup>n</sup>* = −*j*) = 0, (73)

*<sup>n</sup>* = *j*) = *p* �= 0, (74)

. . . *p<sup>λ</sup>* (*s*.*a*ˆ �

*p<sup>λ</sup>* (*s*.*a*ˆ �

*p<sup>λ</sup>* (*s*.*a*ˆ �

end, let us sum all equations in each set. We obtain

. .

. .

*A*1, *A*2,... *An*, and *B* as:

(1 − *pλ*(*s*.*a*ˆ1 = *j*))

(1 − *pλ*(*s*.*a*ˆ*<sup>l</sup>* = *j*))

(1 − *pλ*(*s*.*a*ˆ*<sup>n</sup>* = *j*))

. .

. .

B had a nonzero measure according to the distribution *ρ*, that is, if

. . *p<sup>λ</sup>* (*s*.*a*ˆ �

<sup>1</sup> = *j*)*pλ*(*s*.*a*ˆ

<sup>1</sup> = *j*)*pλ*(*s*.*a*ˆ

<sup>1</sup> = *j*)*pλ*(*s*.*a*ˆ

<sup>1</sup> = *j*)*pλ*(*s*.*a*ˆ

�

�

�

<sup>2</sup> = *j*)... *pλ*(*s*.*a*ˆ

where the first 2*jn* + 1 equations are supposed to hold almost every time within Λ, while the last equation has to be satisfied in a subset of Λ whose measure according to the distribution *ρ*(*λ*) is nonzero. To prove the more general result that no conceivable time-local SHVT can simultaneously satisfy Eqs.(70)-(74), a manipulation of those equations is required. To this

�

 *<sup>p</sup>λ*(*s*. <sup>ˆ</sup> *a* �

 *pλ*(*s*.*a*ˆ �

 *pλ*(*s*.*a*ˆ � belongs to *Al*, then, by definition, *pλ*(*s*.*a*ˆ*<sup>l</sup>* = *j*) = 0, so that Eq.(75) can be satisfied only if *<sup>p</sup>λ*(*s*. <sup>ˆ</sup> *a* � <sup>1</sup> <sup>=</sup> *<sup>j</sup>*)... *<sup>p</sup>λ*(*s*. <sup>ˆ</sup> *a* � *<sup>n</sup>* = *j*) = 0. Hence, for any *λ* ∈ {*A*<sup>1</sup> ∪ *A*<sup>2</sup> ∪ ... ∪ *An*}, we obtain a result leading to a contradiction of Eq.(74), which requires that there is a set of nonzero *ρ* measure within Λ where both probabilities do not vanish. To summarize, we have shown that it is not possible to exhibit any time-local hidden-variable model, satisfying Hardy's logic for *n* successive measurements.

Now, we show that in quantum theory for the *n* successive spin measurement, sometimes *p* > 0. So, we consider *n* successive measurements in directions *s*.*a*ˆ*<sup>i</sup>* (*i* = 1, 2, . . . , *n*) on spin-*s* particles. For a spin-*s* system, we have ( see Appendix-B):

$$|\langle \mathfrak{a}\_{k-1} | \mathfrak{a}\_k \rangle| = |\langle \mathfrak{s}.\mathfrak{a}\_{k-1} | \mathfrak{s}.\mathfrak{a}\_k \rangle| = d\_{\mathfrak{a}\_{k-1}, \mathfrak{a}\_k}^{(s)} (\mathfrak{z}\_k - \mathfrak{z}\_{k-1})\_\prime \tag{78}$$

where *β<sup>k</sup>* is the angle between the *a*ˆ*<sup>k</sup>* and the +*z* axes. So, given the input state |*α*0�, the ( joint) probability that the measurement outcomes will be |*α*1�∈{+*j*,..., −*j*} in the first measurement, |*α*2�∈{+*j*,..., −*j*} in the second measurement,..., |*αn*�∈{+*j*,..., −*j*} in the *n*-th measurement, is given by

$$\begin{split} p(\boldsymbol{a}\_1, \boldsymbol{a}\_2, \dots, \boldsymbol{a}\_n) &= \Pi\_{k=1}^n |\langle \boldsymbol{a}\_{k-1} | \boldsymbol{a}\_k \rangle| \\ &= \Pi\_{k=1}^n d\_{\boldsymbol{a}\_{k-1}, \boldsymbol{a}\_k}^2 (\boldsymbol{\beta}\_k - \boldsymbol{\beta}\_{k-1}). \end{split} \tag{79}$$

We deal with the case where the input state is a pure state whose eigenstates coincide with those of � *S*.*a*ˆ0 for some *a*ˆ0 whose eigenvalues we denote *α*<sup>0</sup> = *j*. Now, by substituting Eq.(79) in the minimal form of Hardy's argument [ Eqs.(64)-(68)], we have

*d* 2 *jj*(*β*1)*d*<sup>2</sup> *jj*(*β*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*1)... *<sup>d</sup>*<sup>2</sup> *jj*(*β<sup>n</sup>* − *<sup>β</sup>n*−1) = 0, (80) *d* 2 *<sup>j</sup>*,*j*−1(*β*1)*d*<sup>2</sup> *<sup>j</sup>*,*j*−1(*<sup>β</sup>* � <sup>2</sup> <sup>−</sup> *<sup>β</sup>*1)... *<sup>d</sup>*<sup>2</sup> *jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0, *d* 2 *<sup>j</sup>*,*j*−2(*β*1)*d*<sup>2</sup> *<sup>j</sup>*,*j*−2(*<sup>β</sup>* � <sup>2</sup> <sup>−</sup> *<sup>β</sup>*1)... *<sup>d</sup>*<sup>2</sup> *jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0, . . *d* 2 *j*,−*j* (*β*1)*d*<sup>2</sup> *j*,−*j* (*β* � <sup>2</sup> <sup>−</sup> *<sup>β</sup>*1)... *<sup>d</sup>*<sup>2</sup> *jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0, . . *d* 2 *jj*(*β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup> *<sup>j</sup>*,*j*−1(*β<sup>l</sup>* <sup>−</sup> *<sup>β</sup>* � *<sup>l</sup>*−1)*d*<sup>2</sup> *<sup>j</sup>*,*j*−1(*<sup>β</sup>* � *<sup>l</sup>*+<sup>1</sup> − *βl*)... *d* 2 *jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0, *d* 2 *jj*(*β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup> *<sup>j</sup>*,*j*−2(*β<sup>l</sup>* <sup>−</sup> *<sup>β</sup>* � *<sup>l</sup>*−1)*d*<sup>2</sup> *<sup>j</sup>*,*j*−2(*<sup>β</sup>* � *<sup>l</sup>*+<sup>1</sup> − *βl*)... *d* 2 *jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0, . . *d* 2 *jj*(*β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup> *j*,−*j* (*β<sup>l</sup>* − *β* � *<sup>l</sup>*−1)*d*<sup>2</sup> *j*,−*j* (*β* � *<sup>l</sup>*+<sup>1</sup> − *βl*)... *d* 2 *jj*(*β* � *<sup>n</sup>* − *β* � *<sup>n</sup>*−1) = 0, (81)

Spin Measurements 25

Quantum Correlations in Successive Spin Measurements 221

� <sup>1</sup> = *π*,

We can obtain this result in the general case. We choose |*β<sup>l</sup>* − *<sup>β</sup>l*−1| = *<sup>π</sup>*, where 2 ≤ *<sup>l</sup>* ≤ *<sup>n</sup>*. Without loss of generality, we select *<sup>β</sup><sup>l</sup>* = *<sup>π</sup>* and *<sup>β</sup>l*−<sup>1</sup> = 0. From the results obtained with

�

We see that *p* > 0 for all spins, and also, the maximum probability of success of Hardy's non-time locality is independent of the number of successive measurements and decreases

*Entanglement in space* displays one of the most interesting features of quantum mechanics, often called quantum non locality. Locality in space and realism impose constraints -Bell's inequalities- on certain combinations of correlations for measurements of spatially separated systems, which are violated by quantum mechanics. Non locality is one of the strangest properties of quantum mechanics, and understanding this notion remains an important

*Entanglement in time* is not introduced in quantum mechanics because of different roles time and space play in quantum theory. The meaning of locality in time is that the results of measurement at time *t*<sup>2</sup> are independent of any measurement performed at some earlier time *t*<sup>1</sup> or later time *t*3. The temporal Bell's inequalities are derived from the realistic hidden

In this chapter we have considered a hidden variable theory of successive measurements on a single spin-*s* system. In all the previous scenarios comparing HVT and QM the principal hypothesis being tested was that, in a given state ( having spatial correlation), HVT implies the existence of a joint probability distribution for all observables even if some of them are not compatible. QM is shown to contradict the consequence of this requirement as it does not assign joint probabilities to the values of incompatible observables. The particular implication that is tested is whether the marginal of the observable *A* in the joint distribution of the compatible observables *A* and *B* is the same as the marginal for *A* in joint distribution

�

*<sup>l</sup>*−<sup>1</sup> <sup>=</sup> *<sup>β</sup><sup>l</sup>* <sup>=</sup> *<sup>π</sup>*. Exactly for (l-1)th in Eq.(84), we have *<sup>β</sup>*

*<sup>l</sup>*−<sup>1</sup> <sup>=</sup> *<sup>π</sup>* and *<sup>β</sup>*

�

1 2 )4*j*

*<sup>l</sup>*−<sup>2</sup> <sup>=</sup> 0 (ii)*<sup>β</sup>*

*p* ≤ ( 1 2 )4*j*

*<sup>l</sup>*+<sup>1</sup> = *π* and *β*

( *β* � *l* <sup>2</sup> ) sin4*<sup>j</sup>* ( *β* � *l* <sup>2</sup> ) <sup>≤</sup> (

( *β* � <sup>3</sup> − *π* <sup>2</sup> )...

<sup>2</sup> ). (86)

. (87)

�

*<sup>l</sup>*−<sup>2</sup> <sup>=</sup> 0. It is easy see that for the first three

� <sup>1</sup> = *β* � �

*<sup>l</sup>*−<sup>1</sup> <sup>=</sup> *<sup>π</sup>* and *<sup>β</sup>*

<sup>2</sup> = ... = *β*

. (88)

*<sup>l</sup>* = *<sup>β</sup>l*−<sup>1</sup> = <sup>0</sup>

� *<sup>l</sup>* = 0

� *<sup>l</sup>*−<sup>1</sup> <sup>=</sup>

(*β*/2), we have

By substituting *d*

By selecting *β*

Eq.(84), *β*

or *β* �

(iii)*β* �

0 and *β* � *<sup>l</sup>*+<sup>2</sup> = *β* �

with *s*.

problem.

variable theory.

�

*<sup>l</sup>*+<sup>1</sup> = *π* and *β*

**7. Summary and comments**

� *<sup>n</sup>* = *β* �

*<sup>l</sup>*+<sup>1</sup> = *β<sup>l</sup>* = *π* or *β*

�

*<sup>l</sup>*+<sup>3</sup> = ... = *β*

*j*

*jj*(*β*) = cos2*<sup>j</sup>*

*p* = cos4*<sup>j</sup>*

cos4*<sup>j</sup>* ( *β* � *<sup>n</sup>* − *β* � *n*−1

*<sup>n</sup>*−<sup>1</sup> <sup>=</sup> ... <sup>=</sup> *<sup>β</sup>*

�

*<sup>l</sup>*−<sup>2</sup> <sup>=</sup> *<sup>β</sup>l*−<sup>1</sup> <sup>=</sup> 0. So we have four cases: (i) *<sup>β</sup>*

*<sup>l</sup>* = 0 and (iv)*β*

�

( *β* � 1 <sup>2</sup> ) cos4*<sup>j</sup>* ( *π* − *β* � 1 <sup>2</sup> ) cos4*<sup>j</sup>*

�

�

cases , the maximum value of *p* is 0, but in the forth case, by selecting *β*

*<sup>n</sup>* = *π*, we get

*p* = cos4*<sup>j</sup>*

<sup>3</sup> = *π* and *β*

$$\begin{cases} d\frac{1}{jj}(\boldsymbol{\beta}\_{1}^{\prime})d\_{jj}^{2}(\boldsymbol{\beta}\_{2}^{\prime}-\boldsymbol{\beta}\_{1}^{\prime})\dots d\_{j,j-1}^{2}(\boldsymbol{\beta}\_{n}-\boldsymbol{\beta}\_{n-1}^{\prime}) = 0, \\ d\frac{1}{jj}(\boldsymbol{\beta}\_{1}^{\prime})d\_{jj}^{2}(\boldsymbol{\beta}\_{2}^{\prime}-\boldsymbol{\beta}\_{1}^{\prime})\dots d\_{j,j-2}^{2}(\boldsymbol{\beta}\_{n}-\boldsymbol{\beta}\_{n-1}^{\prime}) = 0, \\ \vdots \\ d\frac{1}{jj}(\boldsymbol{\beta}\_{1}^{\prime})d\_{jj}^{2}(\boldsymbol{\beta}\_{2}^{\prime}-\boldsymbol{\beta}\_{1}^{\prime})\dots d\_{j,-j}^{2}(\boldsymbol{\beta}\_{n}-\boldsymbol{\beta}\_{n-1}^{\prime}) = 0, \\ d\frac{1}{jj}(\boldsymbol{\beta}\_{1}^{\prime})d\_{jj}^{2}(\boldsymbol{\beta}\_{2}^{\prime}-\boldsymbol{\beta}\_{1}^{\prime})\dots d\_{j,j}^{2}(\boldsymbol{\beta}\_{n}^{\prime}-\boldsymbol{\beta}\_{n-1}^{\prime}) = p. \end{cases} \tag{82}$$

From Eq (80), at least one of the factors must be 0. So

.

$$d\_{jj}^2(\beta\_1) = 0 \Longrightarrow \beta\_1 = \pi$$

$$\text{or}$$

$$d\_{jj}^2(\beta\_2 - \beta\_1) \Longrightarrow |\beta\_2 - \beta\_1| = \pi$$

$$\text{or}$$

$$\cdot$$

$$\cdot$$

$$d\_{jj}^2(\beta\_n - \beta\_{n-1}) \Longrightarrow |\beta\_n - \beta\_{n-1}| = \pi$$

To satisfy all equations (81)-(82), we have the following conditions:

$$\begin{aligned} (\beta\_1 = 0) &\text{ or } (\beta'\_2 = \beta\_1) \\ &\text{and} \\ (\beta\_2 = \beta'\_1) &\text{ or } (\beta\_2 = \beta'\_3) \\ &\cdot \\ &\cdot \\ (\beta\_l = \beta'\_{l-1}) &\text{ or } (\beta\_l = \beta'\_{l+1}) \\ &\cdot \\ &\cdot \\ &\cdot \\ &\text{and} \\ (\beta\_n = \beta'\_{n-1}). \end{aligned} \tag{84}$$

Now, we can calculate the maximum value *p* by using these conditions. For example, if we select *β*<sup>1</sup> = *π*, so we must have *β* � <sup>2</sup> = *β*<sup>1</sup> = *π*. In this case,

$$p = d\_{\vec{j}\vec{j}}^2(\boldsymbol{\beta}\_1^{\prime})d\_{\vec{j}\vec{j}}^2(\pi-\boldsymbol{\beta}\_1^{\prime})d\_{\vec{j}\vec{j}}^2(\boldsymbol{\beta}\_3^{\prime}-\pi)\dots d\_{\vec{j}\vec{j}}^2(\boldsymbol{\beta}\_n^{\prime}-\boldsymbol{\beta}\_{n-1}^{\prime}).\tag{85}$$

By substituting *d j jj*(*β*) = cos2*<sup>j</sup>* (*β*/2), we have

24 Will-be-set-by-IN-TECH

*jj*(*β*1) = 0 =⇒ *β*<sup>1</sup> = *π* or

> or . . .

*jj*(*β*<sup>2</sup> − *β*1) =⇒ |*β*<sup>2</sup> − *β*1| = *π*

*jj*(*β<sup>n</sup>* − *<sup>β</sup>n*−1) =⇒ |*β<sup>n</sup>* − *<sup>β</sup>n*−1| = *<sup>π</sup>*

and

. . .

. . . and (*β<sup>n</sup>* = *β* � *<sup>n</sup>*−1).

Now, we can calculate the maximum value *p* by using these conditions. For example, if we

<sup>3</sup> <sup>−</sup> *<sup>π</sup>*)... *<sup>d</sup>*<sup>2</sup>

*j*,*j* (*β* � *<sup>n</sup>* − *β* �

<sup>2</sup> = *β*<sup>1</sup> = *π*. In this case,

<sup>1</sup>) or (*β*<sup>2</sup> = *β*

*<sup>l</sup>*−1) or (*β<sup>l</sup>* <sup>=</sup> *<sup>β</sup>*

� 3)

�

� <sup>2</sup> = *β*1)

(*β*<sup>1</sup> = 0) or (*β*

(*β*<sup>2</sup> = *β* �

(*β<sup>l</sup>* = *β* �

�

*jj*(*π* − *β* � 1)*d*<sup>2</sup> *jj*(*β* � *<sup>j</sup>*,*j*−1(*β<sup>n</sup>* <sup>−</sup> *<sup>β</sup>*

*<sup>j</sup>*,*j*−2(*β<sup>n</sup>* <sup>−</sup> *<sup>β</sup>*

(*β<sup>n</sup>* − *β* � � *<sup>n</sup>*−1) = 0,

� *<sup>n</sup>*−1) = 0,

*<sup>n</sup>*−1) = 0, (82)

*<sup>n</sup>*−1) = *<sup>p</sup>*. (83)

*<sup>l</sup>*+1) (84)

*<sup>n</sup>*−1). (85)

. . *d* 2 *jj*(*β* � 1)*d*<sup>2</sup> *jj*(*β* � <sup>2</sup> − *β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup>

*d* 2 *jj*(*β* � 1)*d*<sup>2</sup> *jj*(*β* � <sup>2</sup> − *β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup>

*d* 2 *jj*(*β* � 1)*d*<sup>2</sup> *jj*(*β* � <sup>2</sup> − *β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup> *j*,*j* (*β* � *<sup>n</sup>* − *β* �

From Eq (80), at least one of the factors must be 0. So

*d*2

*d*2

*d*2

select *β*<sup>1</sup> = *π*, so we must have *β*

*p* = *d*<sup>2</sup> *jj*(*β* � 1)*d*<sup>2</sup>

To satisfy all equations (81)-(82), we have the following conditions:

. . *d* 2 *jj*(*β* � 1)*d*<sup>2</sup> *jj*(*β* � <sup>2</sup> − *β* � <sup>1</sup>)... *<sup>d</sup>*<sup>2</sup> *j*,−*j*

$$p = \cos^{4j}(\frac{\boldsymbol{\beta}\_1^{\prime}}{2})\cos^{4j}(\frac{\pi-\boldsymbol{\beta}\_1^{\prime}}{2})\cos^{4j}(\frac{\boldsymbol{\beta}\_3^{\prime}-\pi}{2})\dots$$

$$\cos^{4j}(\frac{\boldsymbol{\beta}\_n^{\prime}-\boldsymbol{\beta}\_{n-1}^{\prime}}{2}).\tag{86}$$

By selecting *β* � *<sup>n</sup>* = *β* � *<sup>n</sup>*−<sup>1</sup> <sup>=</sup> ... <sup>=</sup> *<sup>β</sup>* � <sup>3</sup> = *π* and *β* � <sup>1</sup> = *π*,

$$p \le (\frac{1}{2})^{4j}.\tag{87}$$

We can obtain this result in the general case. We choose |*β<sup>l</sup>* − *<sup>β</sup>l*−1| = *<sup>π</sup>*, where 2 ≤ *<sup>l</sup>* ≤ *<sup>n</sup>*. Without loss of generality, we select *<sup>β</sup><sup>l</sup>* = *<sup>π</sup>* and *<sup>β</sup>l*−<sup>1</sup> = 0. From the results obtained with Eq.(84), *β* � *<sup>l</sup>*+<sup>1</sup> = *β<sup>l</sup>* = *π* or *β* � *<sup>l</sup>*−<sup>1</sup> <sup>=</sup> *<sup>β</sup><sup>l</sup>* <sup>=</sup> *<sup>π</sup>*. Exactly for (l-1)th in Eq.(84), we have *<sup>β</sup>* � *<sup>l</sup>* = *<sup>β</sup>l*−<sup>1</sup> = <sup>0</sup> or *β* � *<sup>l</sup>*−<sup>2</sup> <sup>=</sup> *<sup>β</sup>l*−<sup>1</sup> <sup>=</sup> 0. So we have four cases: (i) *<sup>β</sup>* � *<sup>l</sup>*−<sup>1</sup> <sup>=</sup> *<sup>π</sup>* and *<sup>β</sup>* � *<sup>l</sup>*−<sup>2</sup> <sup>=</sup> 0 (ii)*<sup>β</sup>* � *<sup>l</sup>*−<sup>1</sup> <sup>=</sup> *<sup>π</sup>* and *<sup>β</sup>* � *<sup>l</sup>* = 0 (iii)*β* � *<sup>l</sup>*+<sup>1</sup> = *π* and *β* � *<sup>l</sup>* = 0 and (iv)*β* � *<sup>l</sup>*+<sup>1</sup> = *π* and *β* � *<sup>l</sup>*−<sup>2</sup> <sup>=</sup> 0. It is easy see that for the first three cases , the maximum value of *p* is 0, but in the forth case, by selecting *β* � <sup>1</sup> = *β* � <sup>2</sup> = ... = *β* � *<sup>l</sup>*−<sup>1</sup> <sup>=</sup> 0 and *β* � *<sup>l</sup>*+<sup>2</sup> = *β* � *<sup>l</sup>*+<sup>3</sup> = ... = *β* � *<sup>n</sup>* = *π*, we get

$$p = \cos^{4j}(\frac{\beta\_l'}{2})\sin^{4j}(\frac{\beta\_l'}{2}) \le (\frac{1}{2})^{4j}.\tag{88}$$

We see that *p* > 0 for all spins, and also, the maximum probability of success of Hardy's non-time locality is independent of the number of successive measurements and decreases with *s*.

#### **7. Summary and comments**

*Entanglement in space* displays one of the most interesting features of quantum mechanics, often called quantum non locality. Locality in space and realism impose constraints -Bell's inequalities- on certain combinations of correlations for measurements of spatially separated systems, which are violated by quantum mechanics. Non locality is one of the strangest properties of quantum mechanics, and understanding this notion remains an important problem.

*Entanglement in time* is not introduced in quantum mechanics because of different roles time and space play in quantum theory. The meaning of locality in time is that the results of measurement at time *t*<sup>2</sup> are independent of any measurement performed at some earlier time *t*<sup>1</sup> or later time *t*3. The temporal Bell's inequalities are derived from the realistic hidden variable theory.

In this chapter we have considered a hidden variable theory of successive measurements on a single spin-*s* system. In all the previous scenarios comparing HVT and QM the principal hypothesis being tested was that, in a given state ( having spatial correlation), HVT implies the existence of a joint probability distribution for all observables even if some of them are not compatible. QM is shown to contradict the consequence of this requirement as it does not assign joint probabilities to the values of incompatible observables. The particular implication that is tested is whether the marginal of the observable *A* in the joint distribution of the compatible observables *A* and *B* is the same as the marginal for *A* in joint distribution

Spin Measurements 27

Quantum Correlations in Successive Spin Measurements 223

7- We proved that the correlation function between first and third measurement (*t*<sup>1</sup> and *t*3) on spin-*s* particle for a given measurement performed at *t*<sup>2</sup> can not violate the temporal Bell inequality. Therefore, any measurement performed at time *t*<sup>2</sup> disentangles events at time *t*<sup>1</sup>

8- Three successive measurements on spin-*s* particles do not break Svetlinchi Inequality. But it is proved that three successive measurements on qubit violate Scarani-Gisin inequality. Thus, although there are no genuine three-fold temporal correlations, a specific combination of two-fold correlations can have values that are not achievable with correlations in space for

9- Also we showed that three successive measurements violate two types of Bell inequalities involving two and three successive measurements. So two successive measurement correlations are relevant to those of three successive measurements. This behavior is

10- Quantum correlations between two successive measurements on a qubit violates chained Bell inequality which is obtained by providing more than two alternative experiments in every

11- Also, we have studied Hardy's argument for the correlations between the outputs of *n* successive measurements for all *s*-spin measurements. We have shown that the maximum probability of success of Hardy's argument for *n* successive measurements is

<sup>2</sup> )4*s*, which is independent of the number of successive measurements of spin (*n*) and decreases with increase of *s*. This can be compared with the correlations corresponding to measurement of spin observables in a spacelike separated two-particles scenario where only the non-maximally entangled states of any spin-s bipartite system respond to Hardy's

*S* · *a*ˆ1|*a*ˆ0, *α*0� = �*a*ˆ1, *α*0|*e*

where *θ*<sup>1</sup> is the angle between *a*ˆ0 and *a*ˆ1 and *n*ˆ is the unit vector along the direction defined by

*iθ*<sup>1</sup>

*<sup>S</sup>* · *<sup>n</sup>*ˆ, �

*i*� *<sup>S</sup>*·*n*ˆ*θ*<sup>1</sup> (�

<sup>2</sup>*λ*<sup>2</sup> 2!

1! �*a*ˆ1, *<sup>α</sup>*0|[�

*<sup>S</sup>* · *<sup>n</sup>*ˆ, �

*S* · *a*ˆ1)*e*

*S* · *a*ˆ1]|*a*ˆ1, *α*0�

*S* · *a*ˆ1]]|*a*ˆ1*α*0� + ··· (A.3)

*S* · *a*ˆ1|*a*ˆ1, *α*0� = *α*0, (A.4)

<sup>−</sup>*i*�

[*G*, [*G*, *A*]] + ··· (A.2)

*<sup>S</sup>*·*n*ˆ*θ*<sup>1</sup> <sup>|</sup>*a*ˆ1, *<sup>α</sup>*0� (A.1)

We evaluate �*α*1�, �*α*1*α*2� and �*α*1*α*2*α*3� in the state *ρ*<sup>0</sup> given in(4.1).

*iGλAe*−*iG<sup>λ</sup>* <sup>=</sup> *<sup>A</sup>* <sup>+</sup> *<sup>i</sup>λ*[*G*, *<sup>A</sup>*] + *<sup>i</sup>*

*S* · *a*ˆ1|*a*ˆ1, *α*0� +

*<sup>S</sup>* · *<sup>n</sup>*ˆ, [�

�*a*ˆ1, *<sup>α</sup>*0|�

*<sup>α</sup>*<sup>1</sup> *<sup>p</sup>*(*α*1) = �*a*ˆ0, *<sup>α</sup>*0|�

*n*ˆ = *a*ˆ0 × *a*ˆ1. By using Baker- Hausdorff Lemma

�*α*1� <sup>=</sup> �*a*ˆ1, *<sup>α</sup>*0|�

+ *i* <sup>2</sup>*θ*<sup>2</sup> 1 2! �*a*ˆ1, *<sup>α</sup>*0|[�

*e*

are zero.

step.

( 1

(|�

we get,

By using:

nonlocality test.

**8. Appendix A**

�*α*1� =

*S* · *a*ˆ0, *α*0�≡|*a*ˆ0, *α*0�)

*s* ∑ *α*1=−*s*

and *t*<sup>3</sup> if *t*<sup>1</sup> < *t*<sup>2</sup> < *t*3.

any three-qubit system.

analogous to three particle W-state.

for the observables *A* and *C* even if *B* and *C* are not compatible. In other words, HVT implies noncontextuality for which QM can be tested. The celebrated theorem of Bell and Kochen-Specker showed that QM is contextual (Bell, 1966),(Kochen & Specker, 1967). In our scenario, the set of measured observables have a well defined joint probability distribution as each of them acts on a different state. Note that the Bell-type inequalities we have derived follow from equation (4) which says that, for a given value of stochastic hidden variable *λ*, the joint probability for the outcomes of successive measurements must be statistically independent. In other words the hidden variable *λ* completely decides the probabilities of individual measurement outcomes independent of other measurements. We show that QM is not consistent with this requirement of HVT. A Bell-type inequality (for single particle), testing contextuality of QM was proposed by Basu et al. (Basu et al., 2001) and was shown that it could be empirically tested. However, the approach given in the present chapter furnishes a test for realistic nature of QM independent of contextuality. We have compared QM with HVT for different values of spin and for different number of successive measurements. The dependence of the deviation of QM from HVT on the spin value and on the number of successive measurements opens up new possibilities for comparison of these models, and may lead to a sharper understanding of QM.

In the following, I bring some of the key surprising results obtained in this chapter.

1- We obtained temporal Mermin-Klyshko inequality (MKI) and svetlinchi inequality (SI) for *n* successive measurements by using realism and non locality in time. We showed quantum correlations violate temporal MKI and satisfy temporal SI.

2- It was interesting that, for a spin-*s* particle, maximum deviation of quantum mechanics from realism was obtained for all convex combinations of *α*<sup>0</sup> = ±1 states (the case when input state is a mixed state whose eigenstates coincide with those of � *S* · *a*ˆ0 for some *a*ˆ0 whose eigenvalues we denote by *α*<sup>0</sup> ∈ {−*s*,...,*s*}). This is surprising as one would expect pure states to be more 'quantum' than the mixed ones thus breaking Bell inequalities by larger amount.

3- All spin 1/2 states maximally break Mermin-Klyshko inequalities for *n* successive measurements (*η<sup>n</sup>* <sup>=</sup> <sup>√</sup>2) as against only the entangled states break it in multipartite case. Interestingly that for *s* = <sup>1</sup> <sup>2</sup> the random mixture ( maximum noisy state) also breaks BI. This indicates that the notion of "classicality", compatible with the usual local HVT, is different in nature from the notion of classicality that would arise from the non-violation of BI here.

4- We saw that for all spins, BI and MKI is violated in two and three successive measurements (*η*<sup>2</sup> > 1,*η*<sup>3</sup> > 1) and the value of violation MKI in three successive measurements is a little more than the value of violation BI in two successive measurements *η*<sup>3</sup> > *η*<sup>2</sup> except *s* = <sup>1</sup> 2 , while *<sup>η</sup>*<sup>3</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>√</sup>2 for spin *<sup>s</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> . Also *η*<sup>3</sup> and *η*<sup>2</sup> decrease monotonically with increase in the value of *s*. It is interesting, in the case of two and three successive measurements of spin *s* prepared in a pure state, that the maximum violation of BI and MKI falls off as the spin of the particle increases, but tends to a constant for arbitrary large *s*, *η*2(*s* −→ ∞) = 1.143 and *η*3(*s* −→ ∞) = 1.153. It is thus seen that large quantum numbers do not guarantee classical behavior.

5- We showed that for *s* = <sup>1</sup> <sup>2</sup> , the correlation between the outputs of measurements from last *k* out of *n* successive measurements (*k* < *n*) depend on the measurement prior to (*n* − *k*), when *k* is even, while for odd *k*, these correlations are independent of the outputs of measurements prior to *n* − *k*.

6- Interestingly if the initial state is the random mixture or first Stern-Gerloch measurements for the qubit component along the directions *a*<sup>1</sup> perpendicular to initial state *a*ˆ0 ⊥ *a*ˆ1 so that, �*α*1�*QM* = 0, then always quantum averages for all odd number of successive measurements are zero.

26 Will-be-set-by-IN-TECH

for the observables *A* and *C* even if *B* and *C* are not compatible. In other words, HVT implies noncontextuality for which QM can be tested. The celebrated theorem of Bell and Kochen-Specker showed that QM is contextual (Bell, 1966),(Kochen & Specker, 1967). In our scenario, the set of measured observables have a well defined joint probability distribution as each of them acts on a different state. Note that the Bell-type inequalities we have derived follow from equation (4) which says that, for a given value of stochastic hidden variable *λ*, the joint probability for the outcomes of successive measurements must be statistically independent. In other words the hidden variable *λ* completely decides the probabilities of individual measurement outcomes independent of other measurements. We show that QM is not consistent with this requirement of HVT. A Bell-type inequality (for single particle), testing contextuality of QM was proposed by Basu et al. (Basu et al., 2001) and was shown that it could be empirically tested. However, the approach given in the present chapter furnishes a test for realistic nature of QM independent of contextuality. We have compared QM with HVT for different values of spin and for different number of successive measurements. The dependence of the deviation of QM from HVT on the spin value and on the number of successive measurements opens up new possibilities for comparison of these models, and

In the following, I bring some of the key surprising results obtained in this chapter.

'quantum' than the mixed ones thus breaking Bell inequalities by larger amount.

1- We obtained temporal Mermin-Klyshko inequality (MKI) and svetlinchi inequality (SI) for *n* successive measurements by using realism and non locality in time. We showed quantum

2- It was interesting that, for a spin-*s* particle, maximum deviation of quantum mechanics from realism was obtained for all convex combinations of *α*<sup>0</sup> = ±1 states (the case when input state

we denote by *α*<sup>0</sup> ∈ {−*s*,...,*s*}). This is surprising as one would expect pure states to be more

3- All spin 1/2 states maximally break Mermin-Klyshko inequalities for *n* successive measurements (*η<sup>n</sup>* <sup>=</sup> <sup>√</sup>2) as against only the entangled states break it in multipartite case.

indicates that the notion of "classicality", compatible with the usual local HVT, is different in nature from the notion of classicality that would arise from the non-violation of BI here. 4- We saw that for all spins, BI and MKI is violated in two and three successive measurements (*η*<sup>2</sup> > 1,*η*<sup>3</sup> > 1) and the value of violation MKI in three successive measurements is a little more than the value of violation BI in two successive measurements *η*<sup>3</sup> > *η*<sup>2</sup> except *s* = <sup>1</sup>

the value of *s*. It is interesting, in the case of two and three successive measurements of spin *s* prepared in a pure state, that the maximum violation of BI and MKI falls off as the spin of the particle increases, but tends to a constant for arbitrary large *s*, *η*2(*s* −→ ∞) = 1.143 and *η*3(*s* −→ ∞) = 1.153. It is thus seen that large quantum numbers do not guarantee classical

out of *n* successive measurements (*k* < *n*) depend on the measurement prior to (*n* − *k*), when *k* is even, while for odd *k*, these correlations are independent of the outputs of measurements

6- Interestingly if the initial state is the random mixture or first Stern-Gerloch measurements for the qubit component along the directions *a*<sup>1</sup> perpendicular to initial state *a*ˆ0 ⊥ *a*ˆ1 so that, �*α*1�*QM* = 0, then always quantum averages for all odd number of successive measurements

<sup>2</sup> the random mixture ( maximum noisy state) also breaks BI. This

<sup>2</sup> , the correlation between the outputs of measurements from last *k*

<sup>2</sup> . Also *η*<sup>3</sup> and *η*<sup>2</sup> decrease monotonically with increase in

*S* · *a*ˆ0 for some *a*ˆ0 whose eigenvalues

2 ,

may lead to a sharper understanding of QM.

Interestingly that for *s* = <sup>1</sup>

5- We showed that for *s* = <sup>1</sup>

behavior.

prior to *n* − *k*.

while *<sup>η</sup>*<sup>3</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>√</sup>2 for spin *<sup>s</sup>* <sup>=</sup> <sup>1</sup>

correlations violate temporal MKI and satisfy temporal SI.

is a mixed state whose eigenstates coincide with those of �

7- We proved that the correlation function between first and third measurement (*t*<sup>1</sup> and *t*3) on spin-*s* particle for a given measurement performed at *t*<sup>2</sup> can not violate the temporal Bell inequality. Therefore, any measurement performed at time *t*<sup>2</sup> disentangles events at time *t*<sup>1</sup> and *t*<sup>3</sup> if *t*<sup>1</sup> < *t*<sup>2</sup> < *t*3.

8- Three successive measurements on spin-*s* particles do not break Svetlinchi Inequality. But it is proved that three successive measurements on qubit violate Scarani-Gisin inequality. Thus, although there are no genuine three-fold temporal correlations, a specific combination of two-fold correlations can have values that are not achievable with correlations in space for any three-qubit system.

9- Also we showed that three successive measurements violate two types of Bell inequalities involving two and three successive measurements. So two successive measurement correlations are relevant to those of three successive measurements. This behavior is analogous to three particle W-state.

10- Quantum correlations between two successive measurements on a qubit violates chained Bell inequality which is obtained by providing more than two alternative experiments in every step.

11- Also, we have studied Hardy's argument for the correlations between the outputs of *n* successive measurements for all *s*-spin measurements. We have shown that the maximum probability of success of Hardy's argument for *n* successive measurements is ( 1 <sup>2</sup> )4*s*, which is independent of the number of successive measurements of spin (*n*) and decreases with increase of *s*. This can be compared with the correlations corresponding to measurement of spin observables in a spacelike separated two-particles scenario where only the non-maximally entangled states of any spin-s bipartite system respond to Hardy's nonlocality test.

#### **8. Appendix A**

We evaluate �*α*1�, �*α*1*α*2� and �*α*1*α*2*α*3� in the state *ρ*<sup>0</sup> given in(4.1). (|� *S* · *a*ˆ0, *α*0�≡|*a*ˆ0, *α*0�)

$$\langle a\_1 \rangle = \sum\_{a\_1 = -s}^{s} a\_1 p(a\_1) = \langle \hat{a}\_0, a\_0 | \vec{S} \cdot \hat{a}\_1 | \hat{a}\_0, a\_0 \rangle = \langle \hat{a}\_1, a\_0 | e^{i\vec{S} \cdot \hat{\mathcal{H}} \mathbf{e}\_1} (\vec{S} \cdot \hat{a}\_1) e^{-i\vec{S} \cdot \hat{\mathcal{H}} \mathbf{e}\_1} | \hat{a}\_1, a\_0 \rangle \tag{A.1}$$

where *θ*<sup>1</sup> is the angle between *a*ˆ0 and *a*ˆ1 and *n*ˆ is the unit vector along the direction defined by *n*ˆ = *a*ˆ0 × *a*ˆ1. By using Baker- Hausdorff Lemma

$$e^{i\mathbf{G}\lambda}Ae^{-i\mathbf{G}\lambda} = A + i\lambda[\mathbf{G},A] + \left(\frac{i^2\lambda^2}{2!}\right)[\mathbf{G},[\mathbf{G},A]] + \cdots \tag{\text{A.2}}$$

we get,

$$
\begin{split}
\langle\mathfrak{a}\_{1}\rangle &= \langle\mathfrak{a}\_{1},\mathfrak{a}\_{0}|\vec{S}\cdot\mathfrak{A}\_{1}|\mathfrak{a}\_{1},\mathfrak{a}\_{0}\rangle + \frac{i\theta\_{1}}{1!} \langle\mathfrak{a}\_{1},\mathfrak{a}\_{0}|[\vec{S}\cdot\mathfrak{A}\_{\prime}\vec{S}\cdot\mathfrak{A}\_{1}]|\mathfrak{a}\_{1},\mathfrak{a}\_{0}\rangle \\ &+ \frac{i^{2}\theta\_{1}^{2}}{2!} \langle\mathfrak{a}\_{1},\mathfrak{a}\_{0}|[\vec{S}\cdot\mathfrak{A}\_{\prime}[\vec{S}\cdot\mathfrak{A}\_{\prime}\vec{S}\cdot\mathfrak{A}\_{1}]]|\mathfrak{A}\_{1}\mathfrak{a}\_{0}\rangle + \cdots \end{split} \tag{A.3}
$$

By using:

$$<\langle \mathfrak{a}\_{1\prime} \mathfrak{a}\_0 | \vec{S} \cdot \mathfrak{a}\_1 | \mathfrak{a}\_{1\prime} \mathfrak{a}\_0 \rangle = \mathfrak{a}\_{0\prime} \tag{A.4}$$

Spin Measurements 29

Quantum Correlations in Successive Spin Measurements 225

*<sup>S</sup>* · *<sup>n</sup>*ˆ, ··· [

*i <sup>S</sup>*·*n*<sup>ˆ</sup> *<sup>θ</sup>*<sup>1</sup> (

*<sup>S</sup>* · *<sup>n</sup>*ˆ,( *S* · *a*ˆ

*S* · *n*ˆ occurs 2*p* times

[(9*<sup>j</sup>* <sup>−</sup> <sup>1</sup>)*<sup>X</sup>* <sup>−</sup> (9*<sup>j</sup>* <sup>−</sup> <sup>9</sup>)*α*<sup>3</sup>

<sup>0</sup> + *s*(*s* + 1) − 3]

<sup>0</sup> − 3*s*(*s* + 1) + 1].

*<sup>p</sup>α*0*α*0[5*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> <sup>3</sup>*α*<sup>2</sup>

*S* · *n*ˆ occurs odd number of times

8

<sup>0</sup>] ( *<sup>θ</sup>* 2*j* 01 2*j*! ).

<sup>16</sup> cos *<sup>θ</sup>*23{cos *<sup>θ</sup>*1[*<sup>M</sup>* cos<sup>2</sup> *<sup>θ</sup>*<sup>12</sup> <sup>+</sup> *<sup>N</sup>*] + *<sup>R</sup>*[3 cos2 *<sup>θ</sup>*<sup>12</sup> <sup>−</sup> <sup>1</sup>]} (A.19)

<sup>0</sup> + 1]

<sup>0</sup> + *s*(*s* + 1) − 3]

<sup>0</sup> − 3*s*(*s* + 1) + 1].

<sup>0</sup> + 1]

*<sup>S</sup>* · *<sup>a</sup>*ˆ1)3*<sup>e</sup>*

<sup>−</sup>*<sup>i</sup>*

<sup>1</sup>)3]] ··· ]]|*a*ˆ1, *<sup>α</sup>*0�

*<sup>S</sup>*·*n*ˆ*θ*<sup>1</sup> <sup>|</sup>*a*ˆ1, *<sup>α</sup>*0�. (A.15)

)[92*p*−<sup>2</sup> <sup>−</sup> <sup>1</sup>] (A.17)

<sup>0</sup> − 3*s*(*s* + 1) + 1] cos 3*θ*01} (A.18)

(A.16)

<sup>1</sup>|�*a*ˆ0, *<sup>α</sup>*0|*a*ˆ1, *<sup>α</sup>*1�|<sup>2</sup> <sup>=</sup> �*a*ˆ1, *<sup>α</sup>*0|*<sup>e</sup>*

*<sup>S</sup>* · *<sup>n</sup>*ˆ, [

<sup>0</sup>) + *<sup>X</sup>* if

<sup>0</sup> + *α*0(1 − 3*s*(*s* + 1)) *<sup>Y</sup>* <sup>=</sup> <sup>32</sup>*<sup>p</sup>*−<sup>2</sup> <sup>+</sup> <sup>32</sup>*<sup>p</sup>*−<sup>4</sup> <sup>+</sup> ··· <sup>+</sup> <sup>3</sup><sup>2</sup> = ( <sup>9</sup>

where,

where,

we get,

This gives,

where,

*<sup>A</sup>* <sup>=</sup> <sup>1</sup> 8

*<sup>α</sup>*0{[3*α*<sup>2</sup>

�*α*1*α*2*α*3� <sup>=</sup> <sup>1</sup>

If the initial state is a mixed state (4.1),

*<sup>A</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*<sup>1</sup>

Using Baker-Hausdorff lemma and

*α*3

�*a*ˆ1, *<sup>α</sup>*0|[

= ⎧ ⎨ ⎩ *<sup>S</sup>* · *<sup>n</sup>*ˆ, [

*<sup>Y</sup>*(*<sup>X</sup>* <sup>−</sup> *<sup>a</sup>*<sup>3</sup>

*X* = 6*α*<sup>3</sup>

*<sup>A</sup>* <sup>=</sup> <sup>1</sup> 8

After substituting (A.18) in (A.14) and simplifying,

∞ ∑ *j*=0

(−1)*<sup>j</sup>*

<sup>0</sup> <sup>+</sup> <sup>3</sup>*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> <sup>1</sup>] cos *<sup>θ</sup>*<sup>01</sup> + [5*α*<sup>2</sup>

*M* = *α*0[9*α*<sup>2</sup>

*R* = *α*0[5*α*<sup>2</sup>

+*s* ∑ *α*0=−*s*

+*s* ∑ *α*0=−*s*

+*s* ∑ *α*0=−*s*

*M* =

*N* =

*R* =

*<sup>N</sup>* <sup>=</sup> *<sup>α</sup>*0[5*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> <sup>3</sup>*α*<sup>2</sup>

*pα*0*α*0[9*α*<sup>2</sup>

*pα*0*α*0[5*α*<sup>2</sup>

0 if

$$
\langle \hat{a}\_1, a\_0 | [\vec{S} \cdot \hat{n}, \vec{S} \cdot \hat{a}\_1] | \hat{a}\_1, a\_0 \rangle = \langle \hat{a}\_1, a\_0 | (i\vec{S} \cdot (\hat{n} \times \hat{a}\_1)) | \hat{a}\_1, a\_0 \rangle = 0,\tag{A.5}
$$

and

$$
\langle \mathfrak{a}\_{1}, \mathfrak{a}\_{0} | [\overline{\mathcal{S}} \cdot \mathfrak{h}\_{\prime} [\overline{\mathcal{S}} \cdot \mathfrak{h}\_{\prime} \overline{\mathcal{S}} \cdot \mathfrak{h}\_{1}] | \mathfrak{h}\_{1}, \mathfrak{a}\_{0} \rangle = \langle \mathfrak{a}\_{1}, \mathfrak{a}\_{0} | \overline{\mathcal{S}} \cdot \mathfrak{h}\_{1} | \mathfrak{a}\_{1}, \mathfrak{a}\_{0} \rangle = \mathfrak{a}\_{0}.\tag{A.6}
$$

Terms with odd powers of *θ*<sup>1</sup> vanish

$$
\langle \alpha\_1 \rangle = \alpha\_0 - \frac{\theta\_1^2}{2!} \alpha\_0 + \frac{\theta\_1^4}{4!} \alpha\_0 - \dots = \alpha\_0 \cos \theta\_1. \tag{A.7}
$$

If the initial state is mixed state(4.1):

$$<\langle \alpha\_1 \rangle = \sum\_{\alpha\_0 = -s}^{+s} p\_{\mathbb{A}\_0} \alpha\_0 \cos \theta\_1. \tag{A.8}$$

Further we compute

$$
\langle \alpha\_1 \alpha\_2 \rangle = \sum\_{\alpha\_1} \alpha\_1 |\langle \pounds\_0 \alpha\_0 | \pounds\_1 \alpha\_1 \rangle|^2 \sum\_{\mathfrak{a}\_2} \alpha\_2 |\langle \pounds\_1 \alpha\_1 | \pounds\_2 \alpha\_2 \rangle|^2.
$$

By using (A.7)

$$\langle \mathfrak{a}\_{1} \mathfrak{a}\_{2} \rangle = \cos \theta\_{12} \sum\_{\mathfrak{a}\_{1}} \mathfrak{a}\_{1}^{2} |\langle \mathfrak{a}\_{0}, \mathfrak{a}\_{0} | \mathfrak{a}\_{1}, \mathfrak{a}\_{1} \rangle|^{2} = \cos \theta\_{12} \langle \mathfrak{a}\_{0}, \mathfrak{a}\_{0} | (\vec{S} \cdot \mathfrak{a}\_{1})^{2} | \mathfrak{a}\_{0}, \mathfrak{a}\_{0} \rangle$$

$$= \cos \theta\_{12} \langle \mathfrak{a}\_{1}, \mathfrak{a}\_{0} | e^{i \vec{S} \cdot \mathfrak{A} \theta\_{1}} (\vec{S} \cdot \mathfrak{a}\_{1})^{2} e^{-i \vec{S} \cdot \mathfrak{A} \theta\_{1}} | \mathfrak{a}\_{1}, \mathfrak{a}\_{0} \rangle\tag{A.9}$$

Using the Baker-Hausdorff Lemma, and using

$$\langle \hat{a}\_1, a\_0 | [\vec{S} \cdot \hat{n}\_\prime [\vec{S} \cdot \hat{n}\_\prime [\vec{S} \cdot \hat{n}\_\prime \cdots [\vec{S} \cdot \hat{n}\_\prime (\vec{S} \cdot \hat{a}\_1)^2]] \cdot \cdots ] | [\hat{a}\_1, a\_0 ] \tag{A.10}$$

$$= \begin{cases} 0 & \text{if } \vec{S} \cdot \hat{n} \text{ occurs odd number of times} \\\\ 3a\_0^2 - s^2 - s & \text{if } \vec{S} \cdot \hat{n} \text{ occurs } 2p \text{ times} \end{cases}$$

we get,

$$
\langle a\_1 a\_2 \rangle = \frac{1}{2} \cos \theta\_{12} [(s^2 + s - a\_0^2) + (3a\_0^2 - s^2 - s) \cos^2 \theta\_1]. \tag{A.11}
$$

If the initial state is mixed state (4.1),

$$\langle a\_1 a\_2 \rangle = \frac{1}{2} \cos \theta\_{12} \sum\_{a\_0 = -s}^{+s} p\_{a\_0} [(s^2 + s - a\_0^2) + (3a\_0^2 - s^2 - s) \cos^2 \theta\_1]. \tag{A.12}$$

Next we calculate,

$$
\langle a\_1 a\_2 a\_3 \rangle = \sum\_{\mathfrak{a}\_1} a\_1 |\langle \hat{a}\_0, a\_0 | \hat{a}\_1, a\_1 \rangle|^2 \sum\_{\mathfrak{a}\_2} a\_2 |\langle \hat{a}\_1, a\_1 | \hat{a}\_2, a\_2 \rangle|^2 \sum\_{\mathfrak{a}\_3} a\_3 |\langle \hat{a}\_2, a\_2 | \hat{a}\_3, a\_3 \rangle|^2. \tag{A.13}
$$

By using (A.7) and (A.11) we get,

$$
\langle a\_1 a\_2 a\_3 \rangle = \frac{1}{2} a\_0 \cos \theta\_1 \cos \theta\_{23} \sin^2 \theta\_{12} s(s+1) + \frac{1}{2} \cos \theta\_{23} (3 \cos^2 \theta\_{12} - 1) A\_\prime \tag{A.14}
$$

where,

28 Will-be-set-by-IN-TECH

*<sup>S</sup>* · *<sup>a</sup>*ˆ1]]|*a*ˆ1, *<sup>α</sup>*0� <sup>=</sup> �*a*ˆ1, *<sup>α</sup>*0<sup>|</sup>

*θ*4 1

+*s* ∑ *α*0=−*s*

*<sup>α</sup>*1|�*a*ˆ0, *<sup>α</sup>*0|*a*ˆ1, *<sup>α</sup>*1�|<sup>2</sup> <sup>∑</sup>*<sup>α</sup>*<sup>2</sup>

<sup>1</sup>|�*a*ˆ0, *<sup>α</sup>*0|*a*ˆ1, *<sup>α</sup>*1�|<sup>2</sup> <sup>=</sup> cos *<sup>θ</sup>*12�*a*ˆ0, *<sup>α</sup>*0|(

<sup>−</sup>*<sup>i</sup>*

*<sup>S</sup>* · *<sup>n</sup>*ˆ,( *S* · *a*ˆ

*S* · *n*ˆ occurs 2*p* times

0)+(3*α*<sup>2</sup>

1, *α*1|*a*ˆ2, *α*2|

<sup>2</sup> <sup>+</sup> *<sup>s</sup>* <sup>−</sup> *<sup>α</sup>*<sup>2</sup>

*S* · *n*ˆ occurs odd number of times

<sup>0</sup> − *s*

0)+(3*α*<sup>2</sup>

2 ∑*α*3

<sup>0</sup> − *s*

*<sup>S</sup>* · *<sup>a</sup>*ˆ1)2*<sup>e</sup>*

*<sup>S</sup>* · *<sup>n</sup>*ˆ, ··· [

<sup>2</sup> <sup>+</sup> *<sup>s</sup>* <sup>−</sup> *<sup>α</sup>*<sup>2</sup>

*α*2|�*a*ˆ

*pα*<sup>0</sup> [(*s*

*<sup>α</sup>*<sup>0</sup> cos *<sup>θ</sup>*<sup>1</sup> cos *<sup>θ</sup>*<sup>23</sup> sin2 *<sup>θ</sup>*12*s*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) + <sup>1</sup>

*S* · (*n*ˆ × *a*ˆ1))|*a*ˆ1, *α*0� = 0, (A.5)

4! *<sup>α</sup>*<sup>0</sup> −··· <sup>=</sup> *<sup>α</sup>*<sup>0</sup> cos *<sup>θ</sup>*1. (A.7)

*pα*0*α*<sup>0</sup> cos *θ*1. (A.8)

*<sup>S</sup>* · *<sup>a</sup>*ˆ1)2|*a*ˆ0, *<sup>α</sup>*0�

<sup>1</sup>)2]] ··· ]]|*a*ˆ1, *<sup>α</sup>*0� (A.10)

<sup>2</sup> <sup>−</sup> *<sup>s</sup>*) cos<sup>2</sup> *<sup>θ</sup>*1]. (A.11)

<sup>2</sup> <sup>−</sup> *<sup>s</sup>*) cos2 *<sup>θ</sup>*1]. (A.12)

*<sup>α</sup>*3|�*a*ˆ2, *<sup>α</sup>*2|*a*ˆ3, *<sup>α</sup>*3�|2. (A.13)

<sup>2</sup> cos *<sup>θ</sup>*23(3 cos2 *<sup>θ</sup>*<sup>12</sup> <sup>−</sup> <sup>1</sup>)*A*, (A.14)

*<sup>S</sup>*·*n*ˆ*θ*<sup>1</sup> <sup>|</sup>*a*ˆ1, *<sup>α</sup>*0� (A.9)

*<sup>α</sup>*2|�*a*ˆ1, *<sup>α</sup>*1|*a*ˆ2, *<sup>α</sup>*2�|2.

*S* · *a*ˆ1|*a*ˆ1, *α*0� = *α*0. (A.6)

*<sup>S</sup>* · *<sup>a</sup>*ˆ1]|*a*ˆ1, *<sup>α</sup>*0� <sup>=</sup> �*a*ˆ1, *<sup>α</sup>*0|(*<sup>i</sup>*

1 2! *<sup>α</sup>*<sup>0</sup> <sup>+</sup>

�*α*1� =

�*a*ˆ1, *<sup>α</sup>*0|[

�*a*ˆ1, *<sup>α</sup>*0|[

Terms with odd powers of *θ*<sup>1</sup> vanish

If the initial state is mixed state(4.1):

Further we compute

By using (A.7)

we get,

Next we calculate,

�*α*1*α*2*α*3� <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*<sup>1</sup>

�*α*1*α*2*α*3� <sup>=</sup> <sup>1</sup>

By using (A.7) and (A.11) we get,

2

and

*<sup>S</sup>* · *<sup>n</sup>*ˆ,

*<sup>S</sup>* · *<sup>n</sup>*ˆ, [

�*α*1*α*2� <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*<sup>1</sup>

= cos *θ*12�*a*ˆ1, *α*0|*e*

*<sup>S</sup>* · *<sup>n</sup>*ˆ, [

0 if

�*α*1*α*2� <sup>=</sup> cos *<sup>θ</sup>*<sup>12</sup> <sup>∑</sup>*<sup>α</sup>*<sup>1</sup>

Using the Baker-Hausdorff Lemma, and using

�*a*ˆ1, *<sup>α</sup>*0|[

3*α*<sup>2</sup>

<sup>2</sup> cos *<sup>θ</sup>*<sup>12</sup>

*<sup>α</sup>*1|�*a*ˆ0, *<sup>α</sup>*0|*a*ˆ1, *<sup>α</sup>*1�|<sup>2</sup> <sup>∑</sup>*<sup>α</sup>*<sup>2</sup>

�*α*1*α*2� <sup>=</sup> <sup>1</sup>

= ⎧ ⎨ ⎩

If the initial state is mixed state (4.1),

�*α*1*α*2� <sup>=</sup> <sup>1</sup>

*<sup>S</sup>* · *<sup>n</sup>*ˆ,

�*α*1� <sup>=</sup> *<sup>α</sup>*<sup>0</sup> <sup>−</sup> *<sup>θ</sup>*<sup>2</sup>

*α*2

*i <sup>S</sup>*·*n*ˆ*θ*<sup>1</sup> (

*<sup>S</sup>* · *<sup>n</sup>*ˆ, [

<sup>0</sup> <sup>−</sup> *<sup>s</sup>*<sup>2</sup> <sup>−</sup> *<sup>s</sup>* if

<sup>2</sup> cos *<sup>θ</sup>*12[(*<sup>s</sup>*

+*s* ∑ *α*0=−*s*

$$A = \sum\_{\mathfrak{a}\_1} \mathfrak{a}\_1^3 |\langle \mathfrak{a}\_0, \mathfrak{a}\_0 | \mathfrak{a}\_1, \mathfrak{a}\_1 \rangle|^2 = \langle \mathfrak{a}\_1, \mathfrak{a}\_0 | e^{i\vec{\mathcal{S}} \cdot \hbar \theta\_1} (\vec{\mathcal{S}} \cdot \mathfrak{a}\_1)^3 e^{-i\vec{\mathcal{S}} \cdot \hbar \theta\_1} |\mathfrak{a}\_1, \mathfrak{a}\_0 \rangle. \tag{A.15}$$

Using Baker-Hausdorff lemma and

$$= \begin{cases} \langle \boldsymbol{\hbar}\_{1}, \boldsymbol{\alpha}\_{0} \rangle [\boldsymbol{\vec{S}} \cdot \boldsymbol{\hbar}, [\boldsymbol{\vec{S}} \cdot \boldsymbol{\hbar}, [\boldsymbol{\vec{S}} \cdot \boldsymbol{\hbar}, \dots \cdot [\boldsymbol{\vec{S}} \cdot \boldsymbol{\hbar}, (\boldsymbol{\vec{S}} \cdot \boldsymbol{\hbar}\_{1})^{3}]] \cdot \cdots ] ] | \boldsymbol{\hbar}\_{1}, \boldsymbol{\mu}\_{0} \rangle \\\\ \boldsymbol{\hbar} = \begin{cases} \boldsymbol{0} & \text{if } \boldsymbol{\vec{S}} \cdot \boldsymbol{\hbar} \text{ occurs odd number of times} \\\\ \boldsymbol{Y}(\boldsymbol{X} - \boldsymbol{a}\_{0}^{3}) + \boldsymbol{X} & \text{if } \boldsymbol{\vec{S}} \cdot \boldsymbol{\hbar} \text{ occurs } 2p \text{ times} \end{cases} \end{cases} \tag{A.16}$$

where,

$$X = 6a\_0^3 + a\_0(1 - 3s(s+1))$$

$$Y = 3^{2p-2} + 3^{2p-4} + \dots + 3^2 = (\frac{9}{8})[9^{2p-2} - 1] \tag{A.17}$$

we get,

$$A = \frac{1}{8} \sum\_{j=0}^{\infty} (-1)^j [(\Theta^j - 1)X - (\Theta^j - 9)\alpha\_0^3] \left(\frac{\theta\_{01}^{2j}}{2j!} \right).$$

This gives,

$$A = \frac{1}{8}a\_0\{ [3a\_0^2 + 3s(s+1) - 1] \cos \theta\_{01} + [5a\_0^2 - 3s(s+1) + 1] \cos 3\theta\_{01} \} \tag{A.18}$$

After substituting (A.18) in (A.14) and simplifying,

$$\langle a\_1 a\_2 a\_3 \rangle = \frac{1}{16} \cos \theta\_{23} \{ \cos \theta\_1 [M \cos^2 \theta\_{12} + N] + R[3 \cos^2 \theta\_{12} - 1] \} \tag{A.19}$$

where,

$$\begin{aligned} M &= \mathfrak{a}\_0[9\mathfrak{a}\_0^2 + \mathfrak{s}(\mathfrak{s}+1) - 3] \\ N &= \mathfrak{a}\_0[5\mathfrak{s}(\mathfrak{s}+1) - 3\mathfrak{a}\_0^2 + 1] \\ R &= \mathfrak{a}\_0[5\mathfrak{a}\_0^2 - 3\mathfrak{s}(\mathfrak{s}+1) + 1] .\end{aligned}$$

If the initial state is a mixed state (4.1),

$$\begin{aligned} M &= \sum\_{\mathfrak{a}\_0=-s}^{+s} p\_{\mathfrak{a}\_0} \mathfrak{a}\_0 [9\mathfrak{a}\_0^2 + s(s+1) - 3] \\ N &= \sum\_{\mathfrak{a}\_0=-s}^{+s} p\_{\mathfrak{a}\_0} \mathfrak{a}\_0 [5s(s+1) - 3\mathfrak{a}\_0^2 + 1] \\ R &= \sum\_{\mathfrak{a}\_0=-s}^{+s} p\_{\mathfrak{a}\_0} \mathfrak{a}\_0 [5\mathfrak{a}\_0^2 - 3s(s+1) + 1]. \end{aligned}$$

Spin Measurements 31

Quantum Correlations in Successive Spin Measurements 227

Alter, O. & Yamamoto, Y. (n.d.). *Quantum Measurement of a Single System*, A Wiley-interscience

Bennett, C. H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J. & Wootters, W. K. (1996).

Brukner, C., Taylor, S., Cheung, S. & Vedral, V. (2004). *The Seventh International Conference*

Collins, D., Gisin, N., Linden, N., Massar, S. & Popescu., S. (2002). *Phys. Rev. Lett.* 88: 040404. Collins, D., Gisin, N., Popescu, S., Roberts, D. & Scarani, V. (2002). *Phys. Rev. Lett* 88: 170405.

Jarrett, J. P. (1984). On the physical significance of the locality conditions in the bell arguments,

Kaszlikowski, D., Gnacinski, P., Zukowski, M., Miklaszewski, W. & Zeilinger, A. (n.d.).

Peres, A. (1993). *Quantum Theory: Concepts and Methods*, Kluwer Academic Publishers.

Redhead, M. (1987). *Incompleteness, Nonlocality, and Realism*, Clarendon Press, Oxford.

Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. (1969). *Phys. Rev. Lett* 23: 880.

Greenberger, D. M., Horne, M. A. & Zeilinger, A. (1990). *Am. J. Phys.* 58: 1131–1143.

*on Quantum Communication Measurement and Computing, Glasgow, United Kingdom*,

Publication John Wiley and sons, INC.

Beck, C. & Graudenz, D. (1992). *Phys. Rev. A* 46: 6265. Belinskii, A. V. & Klyshko, D. N. (1993). *Phys. Usp.* 36: 653.

Braunstein, L. & Caves, C. M. (1990). *Ann. Phys. (NY)* 202: 22.

Anderson, E., Barnett, S. M. & Aspect, A. (2005). *Phys. Rev. A* 72: 042104.

Ballentine, L. E. (1990). *Quantum Mechanics*, prentice Hall, Englewood Cliffs, NJ. Basu, S., Bandyopadhay, S., Kar, G. & Home, D. (2001). *Phys. Lett. A* 279: 284.

**10. References**

Bell, J. S. (1964). *Physics* 1: 195.

pp. 25–29.

Bell, J. S. (1966). *Rev Mod. Phys.* 38: 447.

*Phys. Rev. Lett* 76: 722. Bohm, D. (1952). *Phys. Rev.* 85: 166.

Cabello, A. (2002a). *Phys. Rev. A* 65: 062105. Cabello, A. (2002b). *Phys. Rev. A* 65: 032108.

Fine, A. (1982). *Phys. Rev. Lett.* 48: 291.

Hardy, L. (1992). *Phys. Rev. Lett* 68: 2981. Hardy, L. (1993). *Phys. Rev. Lett.* 71: 1665.

*Nous* 18: 569–89.

*Phys.Rev. Lett.* 85.

Peres, A. (1992). *Phys. Rev. A* 46: 4413.

Popescu, S. (1995). *Phys. Rev. Lett.* 74: 2619.

Gisin, N. & Peres, A. (1992). *Phys. Lett. A* 162. Goldstein, S. (1994). *Phys. Rev. Lett.* 72: 1951.

Kochen, K. & Specker, E. (1967). *J. Math.* 17: 59. Leggett, A. J. & Garg, A. (1985). *Phys. Rev. Lett.* 54: 857.

Mermin, N. D. (1990). *Phys. Rev. Lett* 65: 1838.

Pagonis, C. & Clifton, R. (1992). *Phys. Lett. A* 168: 100. Parasuram, K. & Ghosh, S. (n.d.). *arXiv: quant-ph/ 0903.3020* . Paz, J. P. & Mahler, G. (1993). *Phys. Rev. Lett.* 71: 3235.

Clifton, R. & Niemann, P. (1992). *Phys. Lett. A* 166: 177.

Collins, D. & Popescu, S. (2001). *J. Phys. A :Math. Gen* 34: 6821. Einstein, A., Podoloski, B. & Rosen, N. (1935). *Phys Rev* 47.

Gisin, N. & Bechmann-Pasquinucci, H. (1998). *Phys. Lett. A* 246.

Heywood, P. & Redhead, M. L. (1983). *Found. Phys.* 13: 481.

Also, by using equations (A.12) and (A.13), one obtains:

$$\begin{aligned} \langle a\_1 a\_3 \rangle &= \sum\_{a\_1} a\_1 |\langle \hat{a}\_0, a\_0 | \hat{a}\_1, a\_1 \rangle|^2 \sum\_{a\_2} |\langle \hat{a}\_1, a\_1 | \hat{a}\_2, a\_2 \rangle|^2 \sum\_{a\_3} a\_3 |\langle \hat{a}\_2, a\_2 | \hat{a}\_3, a\_3 \rangle|^2 \\ &= \cos \theta\_{32} \langle a\_1 a\_2 \rangle \\ &= \frac{1}{2} \cos \theta\_{32} \cos \theta\_{21} [(s^2 + s - a\_0^2) + (3a\_0^2 - s^2 - s)\cos^2 \theta\_1] \end{aligned} \tag{A.20}$$

and we can obtain:

$$
\langle a\_2 a\_3 \rangle = \sum\_{a\_1} |\langle \p\_0, a\_0 | \p\_1, a\_1 \rangle|^2 \sum\_{a\_2} a\_2 |\langle \p\_1, a\_1 | \p\_2, a\_2 \rangle|^2 \sum\_{a\_3} a\_3 |\langle \p\_2, a\_2 | \p\_3, a\_3 \rangle|^2
$$

$$
= \cos \theta\_{32} \langle a\_2^2 \rangle
$$

$$
= \frac{1}{2} s \cos \theta\_{32} \{(s+1) \sin^2 \theta\_{12} + \frac{1}{2} (3 \cos^2 \theta\_{12} - 1) [1 + (2s - 1) \cos^2 \theta\_1] \}. \tag{A.21}
$$

#### **9. Appendix B**

Let us consider a situation where an ensemble of systems prepared in state *ρ* = |*s*.*a*ˆ0 = *α*<sup>0</sup> >< *s*.*a*ˆ0 = *α*0| at time *t* = 0, is subjected to a measurement of the observable *A*(*t*1) = *s*.*a*ˆ1 at time *t*<sup>1</sup> followed by a measurement of the observable *B*(*t*2) = *s*.*a*ˆ2 at time *t*<sup>2</sup> (*t*<sup>2</sup> > *t*<sup>1</sup> > 0), where we have adopted the Heisenberg picture of time evolution. Further, let us assume that both *A*(*t*1) and *B*(*t*2) have purely discrete spectra. Let {*α*1} = {−*s*, −*s* + 1, . . . ,*s*} and {*α*2} <sup>=</sup> {−*s*, <sup>−</sup>*<sup>s</sup>* <sup>+</sup> 1, . . . ,*s*} denote the eigenvalues and *<sup>P</sup>A*(*t*1)(*α*1) = <sup>|</sup>*s*.*a*ˆ1 <sup>=</sup> *<sup>α</sup>*<sup>1</sup> >< *<sup>s</sup>*.*a*ˆ1 <sup>=</sup> *<sup>α</sup>*2|, *<sup>P</sup>B*(*t*2)(*α*2) = <sup>|</sup>*s*.*a*ˆ2 <sup>=</sup> *<sup>α</sup>*<sup>2</sup> >< *<sup>s</sup>*.*a*ˆ2 <sup>=</sup> *<sup>α</sup>*2<sup>|</sup> the corresponding eigenprojectors of *<sup>A</sup>*(*t*1) and *<sup>B</sup>*(*t*2) respectively. Then the joint probability that a measurement of *A*(*t*1) yields the outcome *α*<sup>1</sup> and a measurement of *B*(*t*2) yields the outcome *α*<sup>2</sup> is given by

$$\begin{split} \left| \Pr\_{A(t\_1), B(t\_2)}^{\rho} (\mathfrak{a}\_1, \mathfrak{a}\_2) \right. \\ &= \left| \Pr \left[ P^{B(t\_2)} (\mathfrak{a}\_2) P^{A(t\_1)} (\mathfrak{a}\_1) \rho P^{A(t\_1)} (\mathfrak{a}\_1) P^{B(t\_2)} (\mathfrak{a}\_2) \right] \right| \\ &= \left| \left< \mathfrak{s} \mathfrak{A}\_0 | \mathfrak{s} \mathfrak{A}\_1 \rangle \right|^2 \left| \left< \mathfrak{s} \mathfrak{A}\_1 | \mathfrak{s} \mathfrak{A}\_2 \rangle \right|^2 . \end{split} \tag{B.1} \right| $$

We know that <sup>|</sup>*s*.*<sup>a</sup>* <sup>=</sup> *<sup>α</sup>* <sup>&</sup>gt;<sup>=</sup> <sup>∑</sup>+*<sup>s</sup> <sup>m</sup>*=−*<sup>s</sup> <sup>d</sup>* (*s*) *<sup>α</sup>*,*m*(*β*)|*m* > where *d* (*s*) *<sup>α</sup>m*(*β*) ≡ *s*, *α*|*exp*( −*iSy* (*β*) *<sup>h</sup>*¯ )|*s*, *m* and it obtains Wigner's formula (Sakurai, n.d.) and *α<sup>i</sup>* ∈ −*s*,...,*s* and *β<sup>i</sup>* is the angle between the *a*ˆ*<sup>i</sup>* and the *z* axes. In contrast,

$$\begin{split} |\langle s.\hat{a}\_1 | s.\hat{a}\_2 \rangle| &= \sum\_m  \\ &= \sum\_m d\_{a\_1,m}^{\*(s)}(\beta\_1) d\_{a\_2,m'}^{(j)}(\beta\_2) \\ &= d\_{a\_1a\_2}^{(s)}(\beta\_2-\beta\_1). \end{split} \tag{B.2}$$

So, we obtain

$$pr\_{QM}(\mathfrak{a}\_1, \mathfrak{a}\_2) = |d\_{\mathfrak{a}\_0 \mathfrak{a}\_1}^{(s)} (\mathfrak{f}\_1 - \mathfrak{f}\_0)|^2 |d\_{\mathfrak{a}\_1 \mathfrak{a}\_2}^{(s)} (\mathfrak{f}\_2 - \mathfrak{f}\_1)|^2. \tag{\text{B.3}}$$

#### **10. References**

30 Will-be-set-by-IN-TECH


1, *α*1|*a*ˆ2, *α*2|

<sup>2</sup> <sup>+</sup> *<sup>s</sup>* <sup>−</sup> *<sup>α</sup>*<sup>2</sup>

1 2

Let us consider a situation where an ensemble of systems prepared in state *ρ* = |*s*.*a*ˆ0 = *α*<sup>0</sup> >< *s*.*a*ˆ0 = *α*0| at time *t* = 0, is subjected to a measurement of the observable *A*(*t*1) = *s*.*a*ˆ1 at time *t*<sup>1</sup> followed by a measurement of the observable *B*(*t*2) = *s*.*a*ˆ2 at time *t*<sup>2</sup> (*t*<sup>2</sup> > *t*<sup>1</sup> > 0), where we have adopted the Heisenberg picture of time evolution. Further, let us assume that both *A*(*t*1) and *B*(*t*2) have purely discrete spectra. Let {*α*1} = {−*s*, −*s* + 1, . . . ,*s*} and {*α*2} <sup>=</sup> {−*s*, <sup>−</sup>*<sup>s</sup>* <sup>+</sup> 1, . . . ,*s*} denote the eigenvalues and *<sup>P</sup>A*(*t*1)(*α*1) = <sup>|</sup>*s*.*a*ˆ1 <sup>=</sup> *<sup>α</sup>*<sup>1</sup> >< *<sup>s</sup>*.*a*ˆ1 <sup>=</sup> *<sup>α</sup>*2|, *<sup>P</sup>B*(*t*2)(*α*2) = <sup>|</sup>*s*.*a*ˆ2 <sup>=</sup> *<sup>α</sup>*<sup>2</sup> >< *<sup>s</sup>*.*a*ˆ2 <sup>=</sup> *<sup>α</sup>*2<sup>|</sup> the corresponding eigenprojectors of *<sup>A</sup>*(*t*1) and *<sup>B</sup>*(*t*2) respectively. Then the joint probability that a measurement of *A*(*t*1) yields the outcome *α*<sup>1</sup>

*α*2|�*a*ˆ

1, *α*1|*a*ˆ2, *α*2|

0)+(3*α*<sup>2</sup>

2 ∑*α*3

2 ∑*α*3

*<sup>α</sup>*3|�*a*ˆ2, *<sup>α</sup>*2|*a*ˆ3, *<sup>α</sup>*3�|<sup>2</sup>

(3 cos2 *<sup>θ</sup>*<sup>12</sup> <sup>−</sup> <sup>1</sup>)[<sup>1</sup> + (2*<sup>s</sup>* <sup>−</sup> <sup>1</sup>) cos2 *<sup>θ</sup>*1]}. (A.21)

<sup>0</sup> − *s*

*<sup>α</sup>*3|�*a*ˆ2, *<sup>α</sup>*2|*a*ˆ3, *<sup>α</sup>*3�|<sup>2</sup>

<sup>2</sup> <sup>−</sup> *<sup>s</sup>*) cos<sup>2</sup> *<sup>θ</sup>*1], (A.20)

Also, by using equations (A.12) and (A.13), one obtains:

= cos *θ*32�*α*1*α*2�


*<sup>s</sup>* cos *<sup>θ</sup>*32{(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) sin2 *<sup>θ</sup>*<sup>12</sup> <sup>+</sup>

and a measurement of *B*(*t*2) yields the outcome *α*<sup>2</sup> is given by

*<sup>m</sup>*=−*<sup>s</sup> <sup>d</sup>*


*prQM*(*α*1, *α*2) = |*d*

(*s*)

*m*

= ∑ *m d*

= *d* (*s*)

< *m*|*d*

(*s*)

<sup>∗</sup>(*s*) *<sup>α</sup>*1,*m*(*β*1)*<sup>d</sup>*

*<sup>α</sup>*0*α*<sup>1</sup> (*β*<sup>1</sup> − *β*0)|

(*α*1, *α*2)

(*α*2)*PA*(*t*1)

(*α*1)*ρPA*(*t*1)

*<sup>α</sup>*,*m*(*β*)|*m* > where *d*

it obtains Wigner's formula (Sakurai, n.d.) and *α<sup>i</sup>* ∈ −*s*,...,*s* and *β<sup>i</sup>* is the angle between the

<sup>∗</sup>(*s*) *<sup>α</sup>*1,*m*(*β*1)∑

(*j*) *<sup>α</sup>*2,*m*�(*β*2)

*m*� *d* (*s*)

2|*d* (*s*)

(*α*1)*PB*(*t*2)

<sup>=</sup> |�*s*.*a*ˆ0|*s*.*a*ˆ1�|<sup>2</sup> |�*s*.*a*ˆ1|*s*.*a*ˆ2�|<sup>2</sup> . (B.1)

(*s*) *<sup>α</sup>m*(*β*) ≡ (*α*2) 

*s*, *α*|*exp*(

>

−*iSy* (*β*) *<sup>h</sup>*¯ )|*s*, *m*

2. (B.3)

 and

*α*2,*m*�(*β*2)|*m*�

*<sup>α</sup>*1*α*<sup>2</sup> (*β*<sup>2</sup> − *β*1). (B.2)

*<sup>α</sup>*1*α*<sup>2</sup> (*β*<sup>2</sup> − *β*1)|

*A*(*t*1),*B*(*t*2)

= *Tr PB*(*t*2)

*Pr<sup>ρ</sup>*

We know that <sup>|</sup>*s*.*<sup>a</sup>* <sup>=</sup> *<sup>α</sup>* <sup>&</sup>gt;<sup>=</sup> <sup>∑</sup>+*<sup>s</sup>*

*a*ˆ*<sup>i</sup>* and the *z* axes. In contrast,

So, we obtain

2�

<sup>2</sup> cos *<sup>θ</sup>*<sup>32</sup> cos *<sup>θ</sup>*21[(*<sup>s</sup>*

*<sup>α</sup>*1|�*a*ˆ0, *<sup>α</sup>*0|*a*ˆ1, *<sup>α</sup>*1�|<sup>2</sup> <sup>∑</sup>*<sup>α</sup>*<sup>2</sup>

�*α*1*α*3� <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*<sup>1</sup>

and we can obtain:

**9. Appendix B**

�*α*2*α*3� <sup>=</sup> <sup>∑</sup>*<sup>α</sup>*<sup>1</sup>

<sup>=</sup> <sup>1</sup>

<sup>=</sup> cos *<sup>θ</sup>*32�*α*<sup>2</sup>

<sup>=</sup> <sup>1</sup> 2


**9** 

D. White

*USA* 

**The Gluon Emission** 

*Roosevelt University, Chicago, Illinois* 

**Model for Vector Meson Decay\***

Quantum Chromodynamics (QCD), the heart of the so-called Standard Model, was developed along the lines of the most successful theoretical structure in all of physics, namely, Quantum Electrodynamics (QED), which represents the interactions between the electron and the electromagnetic field, photons serving as the mediator between the two entities. QCD, therefore, contains many objects which are analogous to those within QED. There are the quarks, for example, which come in six "flavors" (up, down, strange, charm, bottom, top), serving as the analogous construct to the electron and its cousins, the muon and the tauon. Plexiformation, of course, always accompanies the proceeding by analogy to any theory, and so we find that quarks must come not only in flavors, but also in "colors" (three total), and they must be fractionally charged (one or two thirds of the electron charge in absolute value). What mediates the quark and the strong field, analogous to the photon in QED, is the gluon. In QED all quantum events are described by a coupling between the electron and the photon, called the fine structure constant of magnitude approximately 0.007. Regarding the coupling between quarks and the strong field responsible for hadronization … the production of hadron pairs in a colliding beams experiment, for example … the situation in QCD is quite different. Work continues at present in the field of high-energy physics to determine the precise nature of the quark-gluon coupling, but one overarching behavior pattern of such coupling, called the strong coupling, is that it is *not* a constant. Rather, it varies generally as the reciprocal of the natural logarithm of the energy

In the work which follows we will have occasion to investigate the phenomenon of vector meson formation and decay in accord with a QCD model, called the Gluon Emission Model (GEM), first developed by F. E. Close in the 1970s. The GEM follows rigorously the precepts of QED proper, the *only* QCD quantity entering into the calculations being the strong coupling parameter, which replaces the fine structure constant in the relevant places. The GEM thus provides for a self-contained formalism that follows the constructs of QED essentially as closely as is possible at the present time. As we will see, even the precise form for the strong coupling parameter may be determined within the GEM, the valid range

Much work presented in this Chapter is taken from D. White, "GEM and the Y(1S)", *The Journal of* 

*Informatics and Mathematical Sciences*, Vol. 2, Nos. 2 & 3 (2010), pp. 71 – 93.

**1. Introduction** 

wrapped up in the colliding beams.

 \*

*Dept. of Biological, Chemical and Physical Sciences,* 

