**3. Bell's inequalities**

The fundamental assumption that the properties of a physical entity are independent and prior to any measurement is called realism. The premise that all physical processes are subject to relativistic causality is called locality. When observations are made at two locations and the only way in which the information of a measurement and it's outcome in the first location can be made available to the second location prior to the measurement made there is through a superluminal signal, we refer to the locations as being located in space-like separated regions. Any theory which asserts that measurements made at space-like separated regions cannot influence outcomes in other regions is called a local theory. All theories where both these conditions of locality and realism are maintained to be valid, are called local realistic theories [23]. Quantum Mechanics is in good part patently non-local and does not uphold realism. We say to a "good part" because separable states in quantum mechanics do not exhibit this property, only entangled states do. While the experimental certification of any local realistic theory or quantum mechanics as the correct description of nature is logically impossible, the consistency of one or the other with observations is feasible within experimental errors. We may test for the compatibility of local or local-realistic theories with experimental facts by supplementing these theories addition assumptions that account for common causes or prior correlations on two systems that had interacted earlier but are now located in space-like separated regions. Such theories are called local hidden variable theories. In trying to establish the appropriateness of a theory, it is customary to look for theories with fewest assumptions and their explanatory power over a wide variety of phenomena. A single failure would of course render it unacceptable. It is within these restrictions that one would look for the contradistinction between a theory or of possible set of theories with others. In the present case, we are interested in the differences in the predictions made by local realistic theories and quantum mechanics.

John Bell [23] proved an extraordinary and significant theorem that imposes quantitative limits on the correlations allowed by local realistic theories. The central result here is that the correlations exhibited by maximally entangled states exceed these limits. Before venturing into Bell discovery, it is first necessary to appreciate that the tests proposed by Bell are not tests on the validity of quantum mechanics per se. Bell tests merely provide an upper bound on the level of correlations that can attained by any local realistic theory. In quantum mechanics, given the state of a multipartite system, the decomposition of such a composite state into product states of the subsystems is in general not possible. For example, there are bi-partite systems j*φ*⟩*AB* that can in general be written as a convex combination of product of the states of the sub-systems: j*ψ*⟩*AB* 6¼ j*φ*⟩*<sup>A</sup>* ⊗ j*χ*⟩*B*. The states that can be so-written are called separable states [2]. States which are not separable are called entangled

#### *Device Independence and the Quest towards Physical Limits of Privacy DOI: http://dx.doi.org/10.5772/intechopen.100364*

states [24]. There are bi-partite states called Bell states that are maximally entangled in that, they exhibit perfect non-local correlations or anti-correlations. The subsystems for such states could either have been generated through a common process of they could have interacted directly or indirectly interacted in the past and may describe a physical property subject to some conservation law. Rather than giving original references we directed the reader to a review article [24] for further details. As an example of Bell states we consider, two photons prepared through the process of spontaneous parametric down conversion (SPDC) [25] that could be prepared in the following Bell states which describe photons entangled in their polarization degree of freedom:

$$|\psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|HH\rangle \pm |VV\rangle)\tag{1}$$

$$|\phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|HV\rangle \pm |VH\rangle)\tag{2}$$

Photons being "flying qubits" each of the two photons could travel to two parties separated by an arbitrary distance. When either of the parties makes a measurement in the V/H basis, both photons in the *ψ*� � � ⟩ state would be found in the horizontal or vertical polarization with equal probability and the states of polarization of the two photons would be perfectly correlated. In case the *ϕ*� � � ⟩, a perfect anti-correlation would be the result. Local measurements performed at spacelike separated regions, ensure the no-signaling condition. The Bell states could very well have been prepared in Bell states in some other basis set such as:

$$|\psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|DD\rangle \pm |AA\rangle)\tag{3}$$

$$|\phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|DA\rangle \pm |AD\rangle)\tag{4}$$

and so forth. In any case, Alice the source of the qubit and Bob the receipient of the qubit could choose to measure their photon in the {H, V} basis or the {D, A} basis. If the Bell states have been prepared with {H, V} polarizations states, only when both Alice and Bob measure with identical basis sets would they end with perfectly correlated or anti-correlated outcomes else their outcomes are perfectly random with respect to each other since {H, V} and {D, A} basis sets used by Alice and Bob for their local measurements are mutually unbiased. The initial states could then be subject to loop-hole-free Bell test by using the two black boxes procured from the supplier. In the case of an entanglement-based implementation of QKD, so long as it is ensured that no classical information such as the measurement outcomes leaks-out from Alice or Bob, the device outputs are secure. The notion of non-locality may best be understood through a Gedankenexperiment popularized by Popescu and Röhrlich [26]. We will first introduce one version of this experiment. The mathematical treatment laid out here closely follows the treatment in [27]. Let Alice and Bob be two stations that are space-like separated. Let these two stations be provided with two black boxes. These boxes have inputs *x* and *y* respectively, where, *x, y* ∈f g 0, 1 . Let these two black boxes be designed to produce outputs *a*, *b* such that *a*, *b* ∈f g �1, þ1 . The **Figure 1**, illustrates this game.

Quite independent of any physical theory, we are free to impose restrictions on the outputs for various possible inputs. The statistical outcomes of such games that can be repeatedly played these boxes is best described through conditional and joint probabilities. We are interested in computing the joint probability the outputs take

**Figure 1.**

*A and B are two black boxes located in space-like separated regions. Inputs x, y to these boxes take value {0, 1}. The output of these boxes a, b assume values {*�*1, +1}. The joint probability p a*ð Þ , *b*j*x*, *y is the quantity of interest in this Gedankenexperiment.*

conditioned by the input setting and their numerical values. Such a joint probability is written as *p a*ð Þ , *b*j*x*, *y* . Here, we have simplified the notation by not writing the setting and the input values separately. When the two boxes are well-separated such that no signal traveling at a finite speed it generally expected that the outputs of the two boxes would be influenced only by the input settings and their values of each box and that the outputs would not be influenced by the input setting of the distant box. This assumption is called a *no-signaling* condition [1]. Under such a constraint, the joint probability would be written as:

$$p(a,b|\varkappa,\boldsymbol{y}) = p(a|\varkappa)\,p(b|\boldsymbol{y}).\tag{5}$$

In writing the joint probability in terms of the product of probabilities as above, the additional assumption is that there are no common causes or past influences that would bring about a correlation between the two boxes. To account for all such possibilities, we may rewrite the conditional probability above as:

$$p(a, b | \mathbf{x}, \mathbf{y}, \boldsymbol{\lambda}) = p(a | \mathbf{x}, \boldsymbol{\lambda}) \cdot p(b | \mathbf{y}, \boldsymbol{\lambda}) \tag{6}$$

where *λ* accounts for all possible common causes and influences. The parameters (s) are sometimes referred to as hidden variables. As an illustrative example of such factors, let us consider two individuals at two distant location who go shopping for a soap dish or a toothbrush and that these two objects generally come in blue or orange color. If a common supplier had supplied a stock of only blue soap dishes and orange toothbrushes, then it should come as no surprises that whenever the two individuals buy the same object they end-up with the same color and whenever they buy different objects, they are of a different color. To factor-in such possibilities, we may rewrite the above joint conditional probability in terms of the product of the individual conditional probabilities. When the Joint conditional probability can be factored as above, we refer to the condition as being non-local. By assuming that the variable λ has a well-defined probability distribution function σ ð Þ*λ* that does depend on the input settings *ia*, *ib* of either of the boxes, we can integrate over that it might take during various runs of the experiment and arrive at

$$p(a,b|\mathbf{x},\mathbf{y}) = \int d\lambda \sigma(\lambda) p(a|\mathbf{x},\lambda) \, p(b|\mathbf{y},\lambda) \tag{7}$$

This condition is then a formal statement of the locality condition. This gist of this statement is that any local operations carried out on either of the two stations oughtn't have any influence on the other station, when the two stations are in

*Device Independence and the Quest towards Physical Limits of Privacy DOI: http://dx.doi.org/10.5772/intechopen.100364*

space-like separated regions. It is implicitly assumed that the choice of input setting is independent of *σ λ*ð Þ which is itself independent of the input settings. In actual implementations, the input settings are chosen with the aid of quantum random number generators. Whenever

$$p(\boldsymbol{x}, \boldsymbol{y}|a, b) \neq p(\boldsymbol{x}|a). p(\boldsymbol{y}|b) \tag{8}$$

the two events are not independent of each other but are correlated. In Bell's original formulation [bell24], he considered only perfect correlations or anticorrelation in the outputs. The CHSH inequality considers the experimentally realistic situation and based on the computation of expectation values of the outputs. Given *x*, *y*∈ f g 0, 1 ∧*a*, *b*∈f g �1, þ1 , the expectation value or the average value over an ensemble of repeated measurement of identically prepared states is given by:

$$
\langle a\_{\mathbf{x}} b\_{\mathbf{y}} \rangle = \sum\_{a,b} ab \ p(ab|\mathbf{x}\mathbf{y}) \tag{9}
$$

Under conditions of objective locality or local realism, the following equality holds:

$$\mathcal{S} = \langle a\_0 b\_0 \rangle + \langle a\_0 b\_1 \rangle + \langle a\_1 b\_0 \rangle - \langle a\_1 b\_1 \rangle \le 2 \tag{10}$$

With quantum systems, S could exceed this value because non-separable states are of a significantly different nature compared local realistic theories. To appreciate this, we may write Bell states in terms of a computational basis as for instance

$$|\psi\rangle = \frac{1}{\sqrt{2}}(|\mathbf{0}0\rangle \pm |\mathbf{1}1\rangle)\tag{11}$$

The vectors 0⟩∧1⟩ are the eigen vectors of the Pauli operator *σz:*Identifying the inputs x and y with measurements along vectorial directions x and y respectively, the quantum mechanical expectation value h*axby*i �*x:y*. When a mutually unbiased basis (MUBS) [28] choice is made for *x, y*, it is trivial to show that *S*≤2 ffiffi 2 <sup>p</sup> .
