**2. Wave and particle debates in QM developments**

#### **2.1 Heisenberg uncertain principle**

The Heisenberg Uncertainty Principle HUP was established in 1927 [Heisenberg (1930)]. The HUP represented a milestone in the early development of quantum theory [Jammer (1974)]. It implies that it is impossible to simultaneously measure the present position of a particle while also determining the future motion of a particle or any system small enough to require a Quantum mechanical treatment. From a mathematical viewpoint, the HUP arises from an equation following the methodology of Fourier analysis for the motion [*Q*, *P*] = *QP* − *PQ* = *ih*¯. The later form of HUP is expressed as �*p* · �*q* ≈ *h*.

This equation shows that the non-commutativity implies that the HUP provides a physical interpretation for the non-commutativity.

#### **2.2 Bohr complementarity principle**

The HUP provided Bohr with a new insight into quantum behaviors [Bohr (1958)]. Bohr established the BCP to extend the idea of complementary variables for the HUP to energy and time, and also to particle and wave behaviors. One must choose between a particle model, with localized positions, trajectories and quanta or a wave model, with spreading wave functions, delocalization and interferences [Jammer (1974)].

Under the BCP, complementary descriptions e.g. wave or particle are mutually exclusive within the same mathematical framework because each model excludes the other. However, a conceptual construction allowed the HUP, the BCP and wave functions together with observed results to be integrated to form the Copenhagen Interpretation of QM. In the context of double slit experiments, the BCP dictates that the observation of an interference pattern for waves and the acquisition of directional information for particles are mutually exclusive.

#### **2.3 Bohr-Einstein debates on wave and particle issues**

Bohr and Einstein remained lifelong friends despite their differences in opinion regarding QM [Bohr (1949; 1958)]. In 1926 Born proposed a probability theory for QM without any causal explanation. Einstein's reaction is well known from his letter to Born [Born (1971)] in which he said "I, at any rate, am convinced that HE [God] does not throw dice."

Then in 1927 at the Solvay Conference, Heisenberg and Bohr announced that the QM revolution was over with nothing further being required. Einstein was dismayed [Bohr (1949); Bolles (2004)] for he believed that the underlying effects were not yet properly understood.

standard wave functions. Simply extending discrete variations using continuous approaches

<sup>343</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

In addition, the following questions need to be addressed in relation to the practical

• Is a pair of complex conjugate objects: *a* + *bi* and *a* − *bi*, a pair of complementary objects? • Is a pair of matrices, a hermit matrix *H* and its complex conjugate matrix *H*∗, a pair of

• Why can density matrix operations on infinite dimension be performed without significant errors while a pair of complementary finite matrices must be restricted by the HUP?

In practice, QM computations are mainly applied to wave functions and [*Q*, *P*] = *ih*¯ formula. Intellectual debate on the theoretical considerations is particularly relevant to the HUP. In comparison, the deeper problems of QM cannot be easily explored in the absence of an experimental approach and a viable alternative theoretical construction [Barrow et al. (2004);

Inspired by the HUP, Bohr uses a continuous analogy and a classical logical construction to describe Quantum systems. The BCP extends the HUP to handle different pairs of opposites

As there were no well refined critical experiments in these days, all the debates between Bohr and Einstein were based on theoretical considerations alone. Compared with Einstein's open-minded attitudes to QM [Einstein et al. (1935)], Bohr and others insisted on the completeness and consistency of the Copenhagen Interpretation on QM [Bohr (1935)]. Such closed attitudes served to distance Bohr and others of like mind from reasonable suggestions made by Einstein and those expressed in the EPR paper and to lead them to treat such suggestions as if they were attacks on their already strongly held views [Bolles (2004)].

The BCP uses a classical logic framework to support dynamic constructions. Underpinned by the BCP, the HUP and a knowledge of wave functions, the Copenhagen Interpretation played a dominant role in QM from the 1930s on as it had by then been accepted as the orthodox

Meanwhile, the EPR paper emphases that critical evidence must be obtained by real experiments and measurements. It is in the nature of a priori philosophical considerations that they will run into difficulties when actual experimental results fail to corresponded with

The EPR position [Einstein et al. (1935)] can be re-visited in the light of modern advances in knowledge and computing theory. From a computing viewpoint, simultaneous properties may be the key with which these long-standing mysteries of QM can at last be unlocked. Operators *P* and *Q* cannot be exchanged, this indicates operational relevances existing in the lower levels of classical QM construction. In addition, there is a requirement of two systems

and to restrict them with exclusive properties [Bohr (1958)].

• What determines when a pair of objects is indeed a complementary pair?

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

presents further difficulties for the HUP.

identification of complementary objects.

complementary objects?

Jammer (1974)].

**3.2 Construction of BCP**

point of view [Jammer (1974)].

their expectations.

**3.3 EPR construction**

Perhaps in a spirit of compromise, Bohr then proposed his BCP that emphases the role of the observer over that which is observed. From 1927-1935, Einstein proposed a series of three intellectual challenges to further explore wave and particle issues [Bohr (1935; 1949); Bolles (2004)]:


#### **2.4 EPR claims**

The key points of the EPR paper are focused on two aspects: either (1) the description of reality given by the wave function in QM is incomplete or (2); the two quantities *P* and *Q* cannot have simultaneous reality.

Both operations: *P* and *Q* are applied *PQ* − *QP* = *ih*¯. Such relationships follow the standard Quantum expression.

Property *PQ* − *QP* �= 0 implies *P* and *Q* operations are related without independent computational properties. Under this condition, it is impossible to execute the two operations simultaneously under extant QM frameworks. From a parallel processing viewpoint, Einstein's view is extremely valuable. As such modern parallel computing theories and practices were only developed in the 1970s [Valiant (1975)] it is remarkable that Einstein pioneered such an approach way back in the 1930s. Modern parallel computing theory and practice support the original EPR paper and the conclusion that a QM description of physical that is expressed only in terms of wave functions is incomplete.

#### **3. Key issues in QM**

#### **3.1 Restriction under HUP**

For the HUP, different interpretations originate from the equation [*Q*, *P*] = *QP* − *PQ* = *ih*¯, and the later HUP form �*q* · �*p* ≥ *h*. From a mathematical viewpoint, this type of inequality implies �*q*, �*p* ≥ *h* too. In other words, a minimal grid of a lattice restricts �*q* and �*p* → 0 actions. From the HUP expression, [*Q*, *P*] �= 0 indicates the construction with a discrete intrinsic limitation. Such structures cannot directly apply to continuous infinitesimal operations.

Many quantum problems do not extend to the region of Plank constant limitations. Investigation back in the 1930s tended to rely more on theoretical considerations rather than actual experimentation [Bohr (1949)]. Consequently many issues had to wait until the 1980s to become better understood.

Both *Q* and *P* are infinite dimensional matrices, the restriction of [*Q*, *P*] = *ih*¯ comes with a clear meaning today on its discrete properties. We cannot apply a continuous approach to make [*Q*, *P*] = 0. From an operational viewpoint, Einstein correctly identified the root of the matter. Since *Q* and *P* cannot exchange, it is not possible to run a simultaneous process on the standard wave functions. Simply extending discrete variations using continuous approaches presents further difficulties for the HUP.

In addition, the following questions need to be addressed in relation to the practical identification of complementary objects.


In practice, QM computations are mainly applied to wave functions and [*Q*, *P*] = *ih*¯ formula. Intellectual debate on the theoretical considerations is particularly relevant to the HUP. In comparison, the deeper problems of QM cannot be easily explored in the absence of an experimental approach and a viable alternative theoretical construction [Barrow et al. (2004); Jammer (1974)].

#### **3.2 Construction of BCP**

4 Measurement Systems

Perhaps in a spirit of compromise, Bohr then proposed his BCP that emphases the role of the observer over that which is observed. From 1927-1935, Einstein proposed a series of three intellectual challenges to further explore wave and particle issues [Bohr (1935; 1949); Bolles

• Third, in 1935 the paper "Can Quantum Mechanical Of Description Physical Reality Be Considered Complete?" [Einstein et al. (1935)] by Einstein, Podolsky and Rosen EPR was

The key points of the EPR paper are focused on two aspects: either (1) the description of reality given by the wave function in QM is incomplete or (2); the two quantities *P* and *Q*

Both operations: *P* and *Q* are applied *PQ* − *QP* = *ih*¯. Such relationships follow the standard

Property *PQ* − *QP* �= 0 implies *P* and *Q* operations are related without independent computational properties. Under this condition, it is impossible to execute the two operations simultaneously under extant QM frameworks. From a parallel processing viewpoint, Einstein's view is extremely valuable. As such modern parallel computing theories and practices were only developed in the 1970s [Valiant (1975)] it is remarkable that Einstein pioneered such an approach way back in the 1930s. Modern parallel computing theory and practice support the original EPR paper and the conclusion that a QM description of physical

For the HUP, different interpretations originate from the equation [*Q*, *P*] = *QP* − *PQ* = *ih*¯, and the later HUP form �*q* · �*p* ≥ *h*. From a mathematical viewpoint, this type of inequality implies �*q*, �*p* ≥ *h* too. In other words, a minimal grid of a lattice restricts �*q* and �*p* → 0 actions. From the HUP expression, [*Q*, *P*] �= 0 indicates the construction with a discrete intrinsic limitation. Such structures cannot directly apply to continuous infinitesimal

Many quantum problems do not extend to the region of Plank constant limitations. Investigation back in the 1930s tended to rely more on theoretical considerations rather than actual experimentation [Bohr (1949)]. Consequently many issues had to wait until the 1980s

Both *Q* and *P* are infinite dimensional matrices, the restriction of [*Q*, *P*] = *ih*¯ comes with a clear meaning today on its discrete properties. We cannot apply a continuous approach to make [*Q*, *P*] = 0. From an operational viewpoint, Einstein correctly identified the root of the matter. Since *Q* and *P* cannot exchange, it is not possible to run a simultaneous process on the

• First, in 1927 Einstein proposed a double slit experiment on interference properties. • Second, in 1930 at the sixth Solvay Congress, Einstein proposed weighing a box emitting

timed releases of electromagnetic radiation.

that is expressed only in terms of wave functions is incomplete.

published in Physical Review.

cannot have simultaneous reality.

(2004)]:

**2.4 EPR claims**

Quantum expression.

**3. Key issues in QM**

operations.

**3.1 Restriction under HUP**

to become better understood.

Inspired by the HUP, Bohr uses a continuous analogy and a classical logical construction to describe Quantum systems. The BCP extends the HUP to handle different pairs of opposites and to restrict them with exclusive properties [Bohr (1958)].

As there were no well refined critical experiments in these days, all the debates between Bohr and Einstein were based on theoretical considerations alone. Compared with Einstein's open-minded attitudes to QM [Einstein et al. (1935)], Bohr and others insisted on the completeness and consistency of the Copenhagen Interpretation on QM [Bohr (1935)]. Such closed attitudes served to distance Bohr and others of like mind from reasonable suggestions made by Einstein and those expressed in the EPR paper and to lead them to treat such suggestions as if they were attacks on their already strongly held views [Bolles (2004)].

The BCP uses a classical logic framework to support dynamic constructions. Underpinned by the BCP, the HUP and a knowledge of wave functions, the Copenhagen Interpretation played a dominant role in QM from the 1930s on as it had by then been accepted as the orthodox point of view [Jammer (1974)].

Meanwhile, the EPR paper emphases that critical evidence must be obtained by real experiments and measurements. It is in the nature of a priori philosophical considerations that they will run into difficulties when actual experimental results fail to corresponded with their expectations.

#### **3.3 EPR construction**

The EPR position [Einstein et al. (1935)] can be re-visited in the light of modern advances in knowledge and computing theory. From a computing viewpoint, simultaneous properties may be the key with which these long-standing mysteries of QM can at last be unlocked. Operators *P* and *Q* cannot be exchanged, this indicates operational relevances existing in the lower levels of classical QM construction. In addition, there is a requirement of two systems

*<sup>X</sup>* <sup>∈</sup> *<sup>B</sup><sup>N</sup>* 2 →

Conditional Meta Measurements CMM

Statistical Distribution SD

*<sup>ρ</sup>*˜ <sup>→</sup> *BP* <sup>→</sup> *<sup>ρ</sup>*˜<sup>−</sup> <sup>→</sup>

→ *ρ*˜<sup>+</sup> →

→ *ρ*˜ →

<sup>345</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

→ {*PH*(*u*˜*β*|*J*)} → → {*PH*(*v*˜*β*|*J*)} →

<sup>∀</sup>*<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>*

(a) Architecture

(b) CIM Component

A comprehensive review of the process of variant construction from conditional probability measurements through to global matrix representations is described briefly in this section. It is hoped that this may offer a convenient path for those seeking to devise and carry out experiments to further explore natural mysteries through the application of sound principles

Using variant principles described in the following subsections, with a *N* bit 0-1 vector *X* and a given logic function *f* , all *N* bit vectors are exhausted, variant measures generate two groups of histograms. The variant simulation and representation system is shown in Fig 2 (a-b). The detailed principles and methods are described in Sections 4.2-4.7 respectively. For multivariate probability conditions, please refer to the chapter of "From local interactive measurements to global matrix representations on variant construction" elsewhere in this book for sample cases

The full measurement and representational architecture as shown in Figure 2(a) has four components: Conditional Meta Measurements CMM, Conditional Interactive Measurements CIM, Statistical Distributions SD and Global Matrix Representations GMR. The key part of the

The Conditional Meta Measurement (CMM) component uses *N* bit 0-1 vector *X* and a given

<sup>2</sup> , CMM transfers *N* bit 0-1 vector under *J*(*X*) to generate four Meta-measures,

Fig. 2. (a-b) Conditional Variant Simulation and Representation System; (a) System

Under this correspondence, Afshar experiments are consistent with the EPR model.

Architecture; (b) Conditional Interactive Measurement CIM Component

**4. Conditional variant simulation and representation system**

and group distributions in multivariate probability environments.

**4.1 Conditional simulation and representation model**

system, the CIM component, is shown in Fig 2(b).

**4.1.1 Conditional Meta Measurements**

function *<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>*

2 →

*SW* <sup>→</sup> *<sup>ρ</sup>*˜−,(<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*˜−)/2 <sup>→</sup> → *ρ*˜+,(1 + *ρ*˜+)/2 →

Conditional Interactive Measurements CIM

> Global Matrix Representations GMR

→ *u*˜ → *v*˜

→ {*M*(*u*˜*β*)} → {*M*(*v*˜*β*)}

*IM* <sup>→</sup> *<sup>u</sup>*˜ → *v*˜

*<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>* 2 →

<sup>∀</sup>*<sup>X</sup>* <sup>∈</sup> *<sup>B</sup><sup>N</sup>*

of logic and measurement.

*u*˜ → *v*˜ →

2 →

$$\begin{array}{c} S\_1 \rightarrow \begin{array}{c} \text{Interactive} \\ \text{Measurements IM} \\ \text{ $IM(S\_1, S\_2 | t), t \in [0, T]} \end{array} \rightarrow \begin{cases} \text{Seperate} \\ \text{Measuresments SM} \\ \text{$ SM(S\_1, S\_2 | t), t > T} \end{cases} \end{array}$$

ERP Model = �*S*1, *<sup>S</sup>*2, *IM*(*S*1, *<sup>S</sup>*2|*t*)*t*∈[0,*T*], *SM*(*S*1, *<sup>S</sup>*2|*t*)*t>T*�

Fig. 1. EPR Measurement Quaternion Model on Einstein's Experimental Devices

to have interactive properties in *t* ∈ [0, *T*] and without interactive properties on *t > T*. Such expressions may not be properly formulated by Fourier transformation schemes on wave functions.

However, after 78 years of development in advanced scienctific and ICT technologies, it is now possible to use advanced photonic and optical fiber technologies to implement all the requirements of the experiments proposed by Einstein.

The core EPR model can be shown in Figure 1, listed notations are explained as follows.

Let *S*<sup>1</sup> be System I, *S*<sup>2</sup> be System II, *IM*(*S*1, *S*2|*t*), *t* ∈ [0, *T*] be Interactive Measurements IM for *S*<sup>1</sup> and *S*<sup>2</sup> on *t* ∈ [0, *T*], *SM*(*S*1, *S*2|*t*), *t > T* be Separate Measurements SM (non-interactive measurements) for *S*<sup>1</sup> and *S*<sup>2</sup> on *t > T*. Einstein's Experimental devices can be described as an EPR measurement quaternion:

$$
\langle \mathcal{S}\_{1\prime} \mathcal{S}\_{2\prime} IM(\mathcal{S}\_{1\prime} \mathcal{S}\_{2} | t)\_{t \in [0, T]} SM(\mathcal{S}\_{1\prime} \mathcal{S}\_{2} | t)\_{t \ge T} \rangle. \tag{2}
$$

If an experiment can be expressed in the requisite form for this model, then it can be legitimately claimed as an EPR experiment.

#### **3.4 Afshar experimental device**

Afshar's experimental results have shown that it is possible to measure both particle and wave interference properties simultaneously in the same experiment with high accuracy [Afshar (2005; 2006); Afshar et al. (2007)]. Since this set of experiments has produced results that challenge the BCP at its very core it is pertinent to analyze and compare the model with the requirements for valid EPR devices.

In Afshar's experiments, {*ψ*1, *ψ*2} are two signals input through double slits; *σ*<sup>1</sup> is the location on the distance *f* to collect interference measurements of {*ψ*1, *ψ*2}, and *σ*<sup>2</sup> is the location on the distance *f* + *d* to collect separate measurements of {*ψ*1, *ψ*2}. Under this configuration, a 1-1 corresponding map can be established as follows:

$$\begin{cases} \psi\_1 \to S\_1; \\ \psi\_2 \to S\_2; \\ \sigma\_1 \to IM(S\_1, S\_2 | t), t \to f; \\ \sigma\_2 \to SM(S\_1, S\_2 | t), t \to f+d. \end{cases} \tag{3}$$

Using quaternion structures,

$$
\langle \psi\_1, \psi\_2, \sigma\_1, \sigma\_2 \rangle \to \langle S\_{1\prime} S\_{2\prime} M(S\_{1\prime} S\_2 | f), SM(S\_{1\prime} S\_2 | f+d) \rangle. \tag{4}
$$

344 Advanced Topics in Measurements From Conditional Probability Measurements to Global Matrix Representations on Variant Construction <sup>7</sup> <sup>345</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

6 Measurement Systems

ERP Model = �*S*1, *<sup>S</sup>*2, *IM*(*S*1, *<sup>S</sup>*2|*t*)*t*∈[0,*T*], *SM*(*S*1, *<sup>S</sup>*2|*t*)*t>T*�

to have interactive properties in *t* ∈ [0, *T*] and without interactive properties on *t > T*. Such expressions may not be properly formulated by Fourier transformation schemes on wave

However, after 78 years of development in advanced scienctific and ICT technologies, it is now possible to use advanced photonic and optical fiber technologies to implement all the

Let *S*<sup>1</sup> be System I, *S*<sup>2</sup> be System II, *IM*(*S*1, *S*2|*t*), *t* ∈ [0, *T*] be Interactive Measurements IM for *S*<sup>1</sup> and *S*<sup>2</sup> on *t* ∈ [0, *T*], *SM*(*S*1, *S*2|*t*), *t > T* be Separate Measurements SM (non-interactive measurements) for *S*<sup>1</sup> and *S*<sup>2</sup> on *t > T*. Einstein's Experimental devices can be described as

If an experiment can be expressed in the requisite form for this model, then it can be

Afshar's experimental results have shown that it is possible to measure both particle and wave interference properties simultaneously in the same experiment with high accuracy [Afshar (2005; 2006); Afshar et al. (2007)]. Since this set of experiments has produced results that challenge the BCP at its very core it is pertinent to analyze and compare the model with the

In Afshar's experiments, {*ψ*1, *ψ*2} are two signals input through double slits; *σ*<sup>1</sup> is the location on the distance *f* to collect interference measurements of {*ψ*1, *ψ*2}, and *σ*<sup>2</sup> is the location on the distance *f* + *d* to collect separate measurements of {*ψ*1, *ψ*2}. Under this configuration, a

> *σ*<sup>1</sup> → *IM*(*S*1, *S*2|*t*), *t* → *f* ; *σ*<sup>2</sup> → *SM*(*S*1, *S*2|*t*), *t* → *f* + *d*.

�*ψ*1, *ψ*2, *σ*1, *σ*2�→�*S*1, *S*2, *IM*(*S*1, *S*2| *f*), *SM*(*S*1, *S*2| *f* + *d*)�. (4)

�*S*1, *<sup>S</sup>*2, *IM*(*S*1, *<sup>S</sup>*2|*t*)*t*∈[0,*T*], *SM*(*S*1, *<sup>S</sup>*2|*t*)*t>T*�. (2)

(3)

The core EPR model can be shown in Figure 1, listed notations are explained as follows.

Fig. 1. EPR Measurement Quaternion Model on Einstein's Experimental Devices

→ {*S*1, *S*2, *t*} →

Separate Measurements SM *SM*(*S*1, *S*2|*t*), *t > T*

*S*<sup>1</sup> →

Interactive Measurements IM *IM*(*S*1, *S*2|*t*), *t* ∈ [0, *T*]

requirements of the experiments proposed by Einstein.

*S*<sup>2</sup> →

an EPR measurement quaternion:

**3.4 Afshar experimental device**

requirements for valid EPR devices.

Using quaternion structures,

1-1 corresponding map can be established as follows:

⎧ ⎪⎪⎨

*ψ*<sup>1</sup> → *S*1; *ψ*<sup>2</sup> → *S*2;

⎪⎪⎩

legitimately claimed as an EPR experiment.

functions.

$$\begin{array}{c} X \in B\_{2}^{N} \to \overline{\{\text{Conditional Met}\}} \quad \begin{array}{c} \text{Conditional Met} \\ \text{Measurements} \\ \text{Messumements} \end{array} \to \bar{\rho} \to \begin{array}{c} \text{Conditional Interactive} \\ \text{Messumements} \\ \text{CIM} \end{array} \to \bar{\upsilon} \end{array}$$

$$\begin{array}{c} \bar{u} \to \overline{\{\text{Statistical}\}} \to \{P\_{H}(\tilde{u}\_{\beta}|I)\} \to \overline{\{\text{Global Matrix}}} \\ \forall \bar{\nu} \to \text{Distribuion} \\ \forall X \in B\_{2}^{N} \to \boxed{\text{SD}} \end{array} \to \begin{array}{c} \{\text{P}\_{H}(\tilde{u}\_{\beta}|I)\} \to \sim \{\text{Global Matrix}\} \\ \text{SD} \end{array} \to \begin{array}{c} \{M(\tilde{u}\_{\beta})\} \to \\ \text{GMR} \end{array} \to \begin{array}{c} \{M(\tilde{u}\_{\beta})\} \to \\ \text{GMR} \end{array} \to \begin{array}{c} \{M(\tilde{u}\_{\beta})\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD}\} \to \sim \{\text{SD$$

(b) CIM Component

Fig. 2. (a-b) Conditional Variant Simulation and Representation System; (a) System Architecture; (b) Conditional Interactive Measurement CIM Component

Under this correspondence, Afshar experiments are consistent with the EPR model.
