**5.3.2 Right: polarized horizontal group**

{*PH*(*u*−|*J*)} elements in Figs 4-5 (b) show (a further) four distinct distributions. Each row contains only one distribution. Sixteen elements in the matrix can be classified into four horizontal classes: {0, 4, 1, 5}, {2, 6, 3, 7}, {8, 12, 9, 13}, {10, 14, 11, 15} respectively. Four meta distributions are given as {0, 2, 8, 10}.

#### **5.3.3 D-P: particle group**

{*PH*(*u*0|*J*)} elements in Figs 4-5 (c) illustrate symmetry properties. There are six pairs of symmetry elements: {8 : 14}, {2 : 11}, {0 : 15}, {6:9}, {4 : 13}, {1:7}. In addition, four elements on anti-diagonals provide different distributions: {10, 12, 3, 5}. Under this condition, ten classes of distributions are distinguished.

#### **5.3.4 D-W: wave group**

{*PH*(*u*1|*J*)} elements in Figs 4-5 (d) illustrate symmetry properties. There are six pairs of symmetry elements: {8 : 14}, {2 : 11}, {0 : 15}, {6:9}, {4 : 13}, {1:7}. In addition, four elements on diagonal positions provide same distribution: {0, 6, 9, 15}. Two elements on anti-diagonals: {12, 3} have the same distribution in Fig 4 (d). Under this condition, nine or ten classes of different distributions can be identified for Fig 4 (d) and Fig 5 (d) respectively.

#### **5.4 Four anti-symmetry groups**

Figures 4-5 (e-h) represent anti-symmetry properties, four groups can be identified.

#### **5.4.1 Left: polarized vertical group**

{*PH*(*v*+|*J*)} elements in Figs 4-5 (e) show that (only) four classes can be distinguished. Elements within these groups members are the same as for symmetry groups in Figs 4-5(a). Their distributions fall within the region [0.5, 1].

**5.7 Polarized effects and double path results**

**5.7.1 Particle distributions and representations**

The equation is true for different values of *N* and *n*.

**5.7.2 Wave distributions and representations**

**5.7.3 Non-symmetry and non-anti-symmetry**

elements can be identified by the relevant variant expressions.

**6. Other relevant measurements and properties**

**6.1 Quaternion measurements**

experiments.

distributions.

measurements.

In order to contrast the different polarized conditions, it is convenient to compare distributions {*PH*(*u*+|*J*), *PH*(*u*−|*J*} and {*PH*(*v*+|*J*), *PH*(*v*−|*J*} arranged according to the corresponding polarized vertical and horizontal effects. This visual effect is similar to what might be found when using polarized filters in order to separate complex signals into two channels. Different

<sup>387</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

For all symmetry or non-symmetry cases under ⊕ asynchronous addition operations, relevant values meet 0 ≤ *u*0, *v*0, *u*−, *v*−, *u*+, *v*<sup>+</sup> ≤ 1. Checking {*PH*(*u*0|*J*), *PH*(*v*0|*J*)} series,

*PH*(*u*0|*J*) = *PH* (*u*−|*J*)+*PH* (*u*+|*J*)

*PH*(*v*0|*J*) = *PH* (*v*−|*J*)+*PH* (*v*+|*J*)

Interference properties are observed in {*PH*(*u*+|*J*) = *PH*(*u*−|*J*)} conditions. Under + synchronous addition operations, relevant values meet 0 ≤ *u*1, *v*1, *u*−, *v*−, *u*+, *v*<sup>+</sup> ≤ 1. Checking {*PH*(*u*1|*J*), *PH*(*v*1|*J*)} distributions and compared with {*PH*(*u*+|*J*), *PH*(*u*−|*J*)} and

*PH*(*u*1|*J*) �<sup>=</sup> *PH*(*u*0|*J*)

Spectra in different cases illustrate wave interference properties. Single and double peaks are shown in interference patterns similar to interference effects in classical double slit

However, for the {*PH*(*u*+|*J*) �= *PH*(*u*−|*J*)} non-symmetry cases, there are significant differences between {*PH*(*u*0|*J*), *PH*(*v*0|*J*)} and {*PH*(*u*1|*J*), *PH*(*v*1|*J*)}. Such cases have interference patterns with more symmetric properties than single path and particle

Four anti-diagonal positions are linked to symmetry and anti-symmetry pairs, twelve other pairs of functions belong to non-symmetry and non-anti-symmetry conditions. Their meta

It is interesting to note the relationship between the variant quaternion and other quaternion

2

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2

*PH*(*v*1|*J*) �<sup>=</sup> *PH*(*v*0|*J*) (14)

distributions can be observed under synchronous and asynchronous conditions.

Construction – A Particle Model of Quantum Interactions for Double Path Experiments

{*PH*(*u*+|*J*), *PH*(*u*−|*J*)} and {*PH*(*v*+|*J*), *PH*(*v*−|*J*)} satisfy following equation.

{*PH*(*v*+|*J*), *PH*(*v*−|*J*)}, non-equations and equations are formulated as follows:

#### **5.4.2 Right: polarized horizontal group**

{*PH*(*v*−|*J*)} elements in Figs 4-5 (f) show that (only) four classes can be distinguished. Elements within these groups are the same as for symmetry groups in Figs 4-5 (b). Their distributions fall within the region [0, 0.5].

#### **5.4.3 D-P: particle group**

{*PH*(*v*0|*J*)} in Figs 4-5 (g) show six pairs of anti-symmetry distributions: {8 14}, {2 11}, {0 15}, {6 9}, {4 13}, {1 7} four elements are distinguished on the anti-diagonals: {10, 12, 3, 5}. Under this condition, ten classes can be identified.

### **5.4.4 D-W: wave group**

{*PH*(*v*1|*J*)} in Figs 4-5 (h) show six pairs of anti-symmetry distributions: {8 14}, {2 11}, {0 15}, {6 9}, {4 13}, {1 7} four pairs of symmetry elements: {3:5}, {10 : 12}, {2:4}, {11 : 13} are distinguished. Under this condition, twelve classes can be identified.

#### **5.5 Odd and even numbers**

From a group view point, only D-P and D-W need to be reviewed as different groups in symmetry conditions. Anti-symmetry conditions are unremarkable.

It is reasonable to suggest that anti-symmetry operations will be much easier to distinguish under experimental conditions, since sixteen groups in D-P conditions and twelve groups in D-W conditions will have significant differences. However, under the symmetry conditions (only) minor differences can be identified.

### **5.5.1 Single and double peaks**

Single and Double peaks can be observed in Fig 4(5) (h): {3, 5} for even and odd numbers respectively.

For two other members {10, 12}, (only) single pulse distributions are observed in Figs 4-5 (h) to show the strongest interference results.

#### **5.6 Class numbers in different conditions**

To summarize over the different classes, 16 matrices are shown in different numbers of identified classes as follows:


where Left:Left Path, Right: Right Path, D-P: Double Path for Particles, D-W: Double Path for Waves; SE: Symmetry for Even number, SO: Symmetry for Odd number, AE: Anti-symmetry for Even number, AO: Anti-symmetry for Odd number.

#### **5.7 Polarized effects and double path results**

16 Measurement Systems

{*PH*(*v*−|*J*)} elements in Figs 4-5 (f) show that (only) four classes can be distinguished. Elements within these groups are the same as for symmetry groups in Figs 4-5 (b). Their

{*PH*(*v*0|*J*)} in Figs 4-5 (g) show six pairs of anti-symmetry distributions: {8 14}, {2 11}, {0 15}, {6 9}, {4 13}, {1 7} four elements are distinguished on the anti-diagonals:

{*PH*(*v*1|*J*)} in Figs 4-5 (h) show six pairs of anti-symmetry distributions: {8 14}, {2 11}, {0 15}, {6 9}, {4 13}, {1 7} four pairs of symmetry elements: {3:5}, {10 : 12}, {2:4}, {11 :

From a group view point, only D-P and D-W need to be reviewed as different groups in

It is reasonable to suggest that anti-symmetry operations will be much easier to distinguish under experimental conditions, since sixteen groups in D-P conditions and twelve groups in D-W conditions will have significant differences. However, under the symmetry conditions

Single and Double peaks can be observed in Fig 4(5) (h): {3, 5} for even and odd numbers

For two other members {10, 12}, (only) single pulse distributions are observed in Figs 4-5 (h)

To summarize over the different classes, 16 matrices are shown in different numbers of

Class No. Left Right D-P D-W SE 4 4 10 9 SO 4 4 10 10 AE 4 4 16 12 AO 4 4 16 12

where Left:Left Path, Right: Right Path, D-P: Double Path for Particles, D-W: Double Path for Waves; SE: Symmetry for Even number, SO: Symmetry for Odd number, AE: Anti-symmetry

13} are distinguished. Under this condition, twelve classes can be identified.

symmetry conditions. Anti-symmetry conditions are unremarkable.

**5.4.2 Right: polarized horizontal group**

distributions fall within the region [0, 0.5].

{10, 12, 3, 5}. Under this condition, ten classes can be identified.

**5.4.3 D-P: particle group**

**5.4.4 D-W: wave group**

**5.5 Odd and even numbers**

**5.5.1 Single and double peaks**

identified classes as follows:

respectively.

(only) minor differences can be identified.

to show the strongest interference results.

**5.6 Class numbers in different conditions**

for Even number, AO: Anti-symmetry for Odd number.

In order to contrast the different polarized conditions, it is convenient to compare distributions {*PH*(*u*+|*J*), *PH*(*u*−|*J*} and {*PH*(*v*+|*J*), *PH*(*v*−|*J*} arranged according to the corresponding polarized vertical and horizontal effects. This visual effect is similar to what might be found when using polarized filters in order to separate complex signals into two channels. Different distributions can be observed under synchronous and asynchronous conditions.

#### **5.7.1 Particle distributions and representations**

For all symmetry or non-symmetry cases under ⊕ asynchronous addition operations, relevant values meet 0 ≤ *u*0, *v*0, *u*−, *v*−, *u*+, *v*<sup>+</sup> ≤ 1. Checking {*PH*(*u*0|*J*), *PH*(*v*0|*J*)} series, {*PH*(*u*+|*J*), *PH*(*u*−|*J*)} and {*PH*(*v*+|*J*), *PH*(*v*−|*J*)} satisfy following equation.

$$\begin{cases} P\_H(u\_0|f) = \frac{P\_H(u\_-|f) + P\_H(u\_+|f)}{2} \\ P\_H(v\_0|f) = \frac{P\_H(v\_-|f) + P\_H(v\_+|f)}{2} \end{cases} \tag{13}$$

The equation is true for different values of *N* and *n*.

#### **5.7.2 Wave distributions and representations**

Interference properties are observed in {*PH*(*u*+|*J*) = *PH*(*u*−|*J*)} conditions. Under + synchronous addition operations, relevant values meet 0 ≤ *u*1, *v*1, *u*−, *v*−, *u*+, *v*<sup>+</sup> ≤ 1. Checking {*PH*(*u*1|*J*), *PH*(*v*1|*J*)} distributions and compared with {*PH*(*u*+|*J*), *PH*(*u*−|*J*)} and {*PH*(*v*+|*J*), *PH*(*v*−|*J*)}, non-equations and equations are formulated as follows:

$$\begin{cases} P\_H(u\_1|J) \neq P\_H(u\_0|J) \\ P\_H(v\_1|J) \neq P\_H(v\_0|J) \end{cases} \tag{14}$$

Spectra in different cases illustrate wave interference properties. Single and double peaks are shown in interference patterns similar to interference effects in classical double slit experiments.

#### **5.7.3 Non-symmetry and non-anti-symmetry**

However, for the {*PH*(*u*+|*J*) �= *PH*(*u*−|*J*)} non-symmetry cases, there are significant differences between {*PH*(*u*0|*J*), *PH*(*v*0|*J*)} and {*PH*(*u*1|*J*), *PH*(*v*1|*J*)}. Such cases have interference patterns with more symmetric properties than single path and particle distributions.

Four anti-diagonal positions are linked to symmetry and anti-symmetry pairs, twelve other pairs of functions belong to non-symmetry and non-anti-symmetry conditions. Their meta elements can be identified by the relevant variant expressions.

#### **6. Other relevant measurements and properties**

#### **6.1 Quaternion measurements**

It is interesting to note the relationship between the variant quaternion and other quaternion measurements.

**6.2.1 Independent conditions in probability**

*P*(*A* ∩ *B*) = *P*(*A*)*P*(*B*)

*P*(*A* ∪ *B*) ≤ *P*(*A*) + *P*(*B*)

For any independent events *A*, *B*,

under specific conditions.

**6.3 Further predictions**

simulations is provided below.

behaviors.

behaviors.

environments.

measurements.

Kolmogorov developed modern probability construction [Ash & Doléans-Dade (2000)] to use measure theory approaches to handle probability measurements. Modern expressions of Bell Inequalities have many forms [SEP (2009)], all of these are based on the conceptual framework of locality which is understood as the conjunction of independent conditions on probability

<sup>389</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Quantum Interactions for Double Path Experiments

*P*(*A* ∪ *B*) = *P*(*A*) + *P*(*B*) − *P*(*A* ∩ *B*), 0 ≤ *P*(*A*), *P*(*B*) ≤ 1

Probability measurement expressions play the core role in Bell Inequalities. In real single

In quantum reality environment, testing measurements could be *<sup>P</sup>*˜(*<sup>A</sup>* <sup>∪</sup> *<sup>B</sup>*) *<sup>&</sup>gt; <sup>P</sup>*˜(*A*) + *<sup>P</sup>*˜(*B*)

From a measurement viewpoint, measurements of local realism correspond to a real number construction that links to Kolmogorov probability [Ash & Doléans-Dade (2000)]. von Neumann (1932,1996)'s mathematical foundation of quantum mechanics is based on a complex number construction. By their nature, these measurement constructions reveal

Probability deductions under local realism must be restricted to real number systems. Under

Observing modern experiments to test Bell Inequations, it is necessary to measure the events in synchronous conditions to create multiple pairs of photons. Different time conditions indicate asynchronous and synchronous conditions playing a critical role in distinguishing between classical and quantum activities. Experimental evidence and case study results are not sufficient at this time to permit firm propositions. However, a summary of predictions for the measurement construction of variant frameworks which can be extrapolated from the

**Prediction 1:** Left distributions have relationships showing polarized vertical behaviors.

**Prediction 2:** Right distributions have relationships showing polarized horizontal behaviors. **Prediction 3:** D-P distributions have relationships showing classical particle statistical

**Prediction 4:** D-W distributions have relationships showing wave interference statistical

**Prediction 5:** Under the same conditions of Bell Inequations, it will be possible to design and implement experiments to distinguish D-P and D-W distributions in real photon experimental

photon experiments, people found that *P*(*A* ∪ *B*) ≤ *P*(*A*) + *P*(*B*) did not hold true.

**6.2.2 Bell inequalities and Newton-Einstein-Feynman particle distributions**

significant differences between the classical and complex probability framework.

the independent condition, *P*(*A* ∪ *B*) ≤ *P*(*A*) + *P*(*B*) is always true.

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#### **6.1.1 Variant quaternion**

In the variant quaternion, <sup>Δ</sup>*f*(*X*)=(*N*⊥, *<sup>N</sup>*+, *<sup>N</sup>*−, *<sup>N</sup>*�), *<sup>N</sup>* = *<sup>N</sup>*<sup>⊥</sup> + *<sup>N</sup>*<sup>+</sup> + *<sup>N</sup>*<sup>−</sup> + *<sup>N</sup>*�.

#### **6.1.2 Einstein quaternion**

Einstein's two-state system of interaction (*N*1, *N*2, *N*12, *N*21) allows the following equations to be established.

$$\begin{cases} N\_1 = N\_\perp + N\_+ \\ N\_2 = N\_- + N\_\top \\ N\_{12} = N\_+ \\ N\_{21} = N\_- \\ N = \quad N\_1 + N\_2 \end{cases} \tag{15}$$

From the equations, the measured pair {*N*21, *N*12} has a 1-1 correspondence to {*N*−, *N*+}.

#### **6.1.3 CHSH quaternion**

Selecting + → 1, − → 0, CHSH's *N*±,±(*a*, *b*) measurements meet

$$\begin{cases} N\_{+,+}(a,b) \to N\_{\top} \\ N\_{+,-}(a,b) \to N\_{-} \\ N\_{-,+}(a,b) \to N\_{+} \\ N\_{-,-}(a,b) \to N\_{\perp} \end{cases} \tag{16}$$

(*N*++, *<sup>N</sup>*+−, *<sup>N</sup>*−+, *<sup>N</sup>*−−) → (*N*�, *<sup>N</sup>*−, *<sup>N</sup>*+, *<sup>N</sup>*⊥), let *<sup>N</sup>* = *<sup>N</sup>*++ + *<sup>N</sup>*+<sup>−</sup> + *<sup>N</sup>*−<sup>+</sup> + *<sup>N</sup>*−−. CHSH quaternion is a permutation of the variant quaternion.

#### **6.1.4 Aspect quaternion**

Aspect's quaternion (*Nt*, *Nr*, *Nc*, *Nω*) have following corresponding:

$$\begin{cases} N\_t \to N\_-\\ N\_r \to N\_+\\ N\_\omega \to N \end{cases} \tag{17}$$

For *Nc*, there is no parameter in the variant quaternion for parameter *Nc*. *Nc* indicates joined action numbers to distinguish single and double paths, corresponding to {*u*1, *v*1} times.

This parameter is of significance in an actual experiment. In a simulated system, the parameter provides a control coefficient that separates two types of paths {*u*0, *v*0} and {*u*1, *v*1} that would be measured in real experiments.

#### **6.2 Different particle models**

From Newton's particles to Young's Double slit experiments, the question of how to distinguish particle and wave measurements has a long history [Hawking & Mlodinow (2010); Penrose (2004)]. From a measurement viewpoint, recent activities testing Bell Inequations can be seen to be consistent with historical viewpoints [Jammer (1974)].

The fundamental assumptions of Bell Inequations are based on a local realism [Eberhard (1978); Fine (1999)]. A key condition of measure theory can be seen in a review of authoritative definitions of local realism [SEP (2009)].

#### **6.2.1 Independent conditions in probability**

Kolmogorov developed modern probability construction [Ash & Doléans-Dade (2000)] to use measure theory approaches to handle probability measurements. Modern expressions of Bell Inequalities have many forms [SEP (2009)], all of these are based on the conceptual framework of locality which is understood as the conjunction of independent conditions on probability measurements.

For any independent events *A*, *B*,

18 Measurement Systems

Einstein's two-state system of interaction (*N*1, *N*2, *N*12, *N*21) allows the following equations to

*<sup>N</sup>*<sup>1</sup> = *<sup>N</sup>*<sup>⊥</sup> + *<sup>N</sup>*<sup>+</sup> *<sup>N</sup>*<sup>2</sup> = *<sup>N</sup>*<sup>−</sup> + *<sup>N</sup>*� *N*<sup>12</sup> = *N*<sup>+</sup> *N*<sup>21</sup> = *N*<sup>−</sup> *N* = *N*<sup>1</sup> + *N*<sup>2</sup>

From the equations, the measured pair {*N*21, *N*12} has a 1-1 correspondence to {*N*−, *N*+}.

*<sup>N</sup>*+,+(*a*, *<sup>b</sup>*) → *<sup>N</sup>*� *N*+,−(*a*, *b*) → *N*<sup>−</sup> *N*−,+(*a*, *b*) → *N*<sup>+</sup> *<sup>N</sup>*−,−(*a*, *<sup>b</sup>*) → *<sup>N</sup>*<sup>⊥</sup>

(*N*++, *<sup>N</sup>*+−, *<sup>N</sup>*−+, *<sup>N</sup>*−−) → (*N*�, *<sup>N</sup>*−, *<sup>N</sup>*+, *<sup>N</sup>*⊥), let *<sup>N</sup>* = *<sup>N</sup>*++ + *<sup>N</sup>*+<sup>−</sup> + *<sup>N</sup>*−<sup>+</sup> + *<sup>N</sup>*−−. CHSH

*Nt* → *N*<sup>−</sup> *Nr* → *N*<sup>+</sup> *N<sup>ω</sup>* → *N*

For *Nc*, there is no parameter in the variant quaternion for parameter *Nc*. *Nc* indicates joined action numbers to distinguish single and double paths, corresponding to {*u*1, *v*1} times. This parameter is of significance in an actual experiment. In a simulated system, the parameter provides a control coefficient that separates two types of paths {*u*0, *v*0} and {*u*1, *v*1} that

From Newton's particles to Young's Double slit experiments, the question of how to distinguish particle and wave measurements has a long history [Hawking & Mlodinow (2010); Penrose (2004)]. From a measurement viewpoint, recent activities testing Bell Inequations can

The fundamental assumptions of Bell Inequations are based on a local realism [Eberhard (1978); Fine (1999)]. A key condition of measure theory can be seen in a review of authoritative

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(16)

(17)

In the variant quaternion, <sup>Δ</sup>*f*(*X*)=(*N*⊥, *<sup>N</sup>*+, *<sup>N</sup>*−, *<sup>N</sup>*�), *<sup>N</sup>* = *<sup>N</sup>*<sup>⊥</sup> + *<sup>N</sup>*<sup>+</sup> + *<sup>N</sup>*<sup>−</sup> + *<sup>N</sup>*�.

⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎧ ⎪⎪⎨

⎪⎪⎩

Aspect's quaternion (*Nt*, *Nr*, *Nc*, *Nω*) have following corresponding:

be seen to be consistent with historical viewpoints [Jammer (1974)].

⎧ ⎨ ⎩

Selecting + → 1, − → 0, CHSH's *N*±,±(*a*, *b*) measurements meet

quaternion is a permutation of the variant quaternion.

**6.1.1 Variant quaternion**

**6.1.2 Einstein quaternion**

**6.1.3 CHSH quaternion**

**6.1.4 Aspect quaternion**

would be measured in real experiments.

definitions of local realism [SEP (2009)].

**6.2 Different particle models**

be established.

$$\begin{array}{l}P(A \cap B) = P(A)P(B) \\ P(A \cup B) = P(A) + P(B) - P(A \cap B), 0 \le P(A), P(B) \le 1 \\ P(A \cup B) \le P(A) + P(B) \end{array} \tag{18}$$

Probability measurement expressions play the core role in Bell Inequalities. In real single photon experiments, people found that *P*(*A* ∪ *B*) ≤ *P*(*A*) + *P*(*B*) did not hold true.

In quantum reality environment, testing measurements could be *<sup>P</sup>*˜(*<sup>A</sup>* <sup>∪</sup> *<sup>B</sup>*) *<sup>&</sup>gt; <sup>P</sup>*˜(*A*) + *<sup>P</sup>*˜(*B*) under specific conditions.

#### **6.2.2 Bell inequalities and Newton-Einstein-Feynman particle distributions**

From a measurement viewpoint, measurements of local realism correspond to a real number construction that links to Kolmogorov probability [Ash & Doléans-Dade (2000)]. von Neumann (1932,1996)'s mathematical foundation of quantum mechanics is based on a complex number construction. By their nature, these measurement constructions reveal significant differences between the classical and complex probability framework.

Probability deductions under local realism must be restricted to real number systems. Under the independent condition, *P*(*A* ∪ *B*) ≤ *P*(*A*) + *P*(*B*) is always true.

#### **6.3 Further predictions**

Observing modern experiments to test Bell Inequations, it is necessary to measure the events in synchronous conditions to create multiple pairs of photons. Different time conditions indicate asynchronous and synchronous conditions playing a critical role in distinguishing between classical and quantum activities. Experimental evidence and case study results are not sufficient at this time to permit firm propositions. However, a summary of predictions for the measurement construction of variant frameworks which can be extrapolated from the simulations is provided below.

**Prediction 1:** Left distributions have relationships showing polarized vertical behaviors.

**Prediction 2:** Right distributions have relationships showing polarized horizontal behaviors.

**Prediction 3:** D-P distributions have relationships showing classical particle statistical behaviors.

**Prediction 4:** D-W distributions have relationships showing wave interference statistical behaviors.

**Prediction 5:** Under the same conditions of Bell Inequations, it will be possible to design and implement experiments to distinguish D-P and D-W distributions in real photon experimental environments.

(c) D-P

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Construction – A Particle Model of Quantum Interactions for Double Path Experiments

(d) D-W

20 Measurement Systems

(a) Left

(b) Right

(g) D-P

<sup>393</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Quantum Interactions for Double Path Experiments

(h) D-W

Representations. (a) Left; (b) Right; (c) D-P; (d)D-W in symmetry conditions; (e) Left; (f)

<sup>2</sup> Eight Matrices of Global Matrix

Fig. 4. (a-h) Even number groups: *<sup>N</sup>* <sup>=</sup> {12}, *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>4</sup>

Right; (g) D-P; (h)D-W in anti-symmetry conditions.

(f) Right

22 Measurement Systems

(e) Left

(f) Right

Fig. 4. (a-h) Even number groups: *<sup>N</sup>* <sup>=</sup> {12}, *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>4</sup> <sup>2</sup> Eight Matrices of Global Matrix Representations. (a) Left; (b) Right; (c) D-P; (d)D-W in symmetry conditions; (e) Left; (f) Right; (g) D-P; (h)D-W in anti-symmetry conditions.

(c) D-P

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Construction – A Particle Model of Quantum Interactions for Double Path Experiments

(d) D-W

24 Measurement Systems

(a) Left

(b) Right

(g) D-P

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Construction – A Particle Model of Quantum Interactions for Double Path Experiments

(h) D-W

Representations. (a) Left; (b) Right; (c) D-P; (d)D-W in symmetry conditions; (e) Left; (f)

<sup>2</sup> Eight Matrices of Global Matrix

Fig. 5. (a-h) Odd number groups: *<sup>N</sup>* <sup>=</sup> {13}, *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>4</sup>

Right; (g) D-P; (h)D-W in anti-symmetry conditions.

(f) Right

26 Measurement Systems

(e) Left

(f) Right

Fig. 5. (a-h) Odd number groups: *<sup>N</sup>* <sup>=</sup> {13}, *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>4</sup> <sup>2</sup> Eight Matrices of Global Matrix Representations. (a) Left; (b) Right; (c) D-P; (d)D-W in symmetry conditions; (e) Left; (f) Right; (g) D-P; (h)D-W in anti-symmetry conditions.

Six predictions and two conjectures are summarized in this chapter to guide further theoretical

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In addition, real world experiments are expected to be designed and implemented in the near

Thanks to Colin W. Campbell for help with the English edition, to The School of Software Engineering, Yunnan University and The Key Laboratory of Yunnan Software Engineering for financial supports to the Information Security research projects (2010EI02, 2010KS06) and

Afshar, S., Flores, E., McDonald, K. & Knoesel, E. (2007). Paradox in wave particle duality,

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**Prediction 6:** It will be much easier to design and implement key experiments to distinguish D-P and D-W behaviors in asynchronous conditions than in synchronous conditions.

In other words, under proposed variant measurements, the simplest effects are polarized properties in Left and Right matrices. Both D-P and D-W distributions are generated from pairs of polarized signals in general cases. In addition, significant differences can be observed between D-P and D-W distributions in asynchronous conditions. This set of theoretical predictions could help experimenters to design and implement effective experiments to check variant measurements under real quantum environments.

#### **6.4 Two conjectures**

Back to Young's waves and Newton's particles, Bohr's complementarity, EPR and Feynman's particle and wave conditions [Hawking & Mlodinow (2010); Jammer (1974); Penrose (2004)], it is essential to list two conjectures to summarize our results as follows:

**Conjecture 1.** Measurement results of Newton-Einstein-Feynman particles and Variant D-P models must obey Bell Inequations.

This conjecture could be approved from listed models satisfied independent conditions. From this viewpoint, Newton-Einstein-Feynman particle models and Variant D-P models could satisfy Bell Inequalities. Bell Inequations at most could provide only a logical foundation for different particle models.

**Conjecture 2.** Measurement results of Young-Bohr-Feynman waves and Variant D-W models satisfy the same types of entanglement conditions.

Since the Local Realism cannot be supported by quantum construction, a solid foundations is required to validate this conjecture using complex-probability conditions for different entanglements in real quantum environments.

#### **7. Conclusion**

Analyzing a *N* bit 0-1 vector and its exhaustive sequences for variant measurement, from a double path experiment viewpoint, this system simulates double path interference properties through different accurate distributions from local interactive measurements to global matrix representations. Using this model, two groups of parameters {*uβ*} and {*vβ*} describe left path, right path, and double paths for particles and double paths for waves with distinguishing symmetry and anti-symmetry properties. {*PH*(*uβ*|*J*), *PH*(*vβ*|*J*)} provide eight groups of distributions under symmetry and anti-symmetry forms. In addition, {*M*(*uβ*), *M*(*vβ*)} provide eight matrices to illustrate global behaviors under complex conditions.

Compared with the variant quaternion and other quaternion measurements, it is helpful to understand the usefulness and limitations of variant simulation properties.

The complexity of *n*-variable function space has a size of 22*<sup>n</sup>* and exhaustive vector space has 2*N*. Whole simulation complexity is determined by *O*(22*<sup>n</sup>* <sup>×</sup> <sup>2</sup>*N*) as ultra exponent productions. How to overcome the limitations imposed by such complexity and how best to compare and contrast such simulations with real world experimentation will be key issues in future work.

Six predictions and two conjectures are summarized in this chapter to guide further theoretical and experimental exploration.

In addition, real world experiments are expected to be designed and implemented in the near future to test results given in this chapter.
