**2. Double path model and measurements of quantum interaction**

#### **2.1 Mach-Zehnder interferometer model**

The Mach-Zehnder interferometer is the most popular device used to support a Young double slit experiment.

In Fig 1(a) a double path interferometer is shown. An input signal *X* under control function *f* causes Laser *LS* to emit the output signal *ρ* under *BP* (Bi-polarized filter) operation. The output is in the form of a pair of signals: *ρ*<sup>+</sup> and *ρ*−. Both signals are processed by *SW* output *ρ*+ *<sup>L</sup>* and *ρ*<sup>−</sup> *<sup>R</sup>* , and then *IM* to generate output signals *IM*(*ρ*<sup>+</sup> *<sup>L</sup>* , *ρ*<sup>−</sup> *<sup>R</sup>* ) . In Fig 1(b), a representation model has been described with the same signals being used.

#### **2.1.1 Other devices**

2 Measurement Systems

Mach-Zehnder interferometers and Stern-Gerlach spin-devices play a key role in Quantum measurement development [Barnett (2009); Barrow et al. (2004); Hawking & Mlodinow (2010); Jammer (1974)]. Wave particle-duality has been demonstrated in larger particles [Arndt et al. (1999)] and advanced optical fibers, communication, computer software, photonics, and integrated technologies have been applied to different quantum media [Barrow et al. (2004);

In the 1960s, Bell played an important role in exploring the foundations of the quantum approach [Bell (1964)]. Based on the EPR paradox, he proposed inequations for measurable experiments to distinguish between Bohr's Principle of Complementarity and Einstein's EPR

By the 1970s, work piloted by [Clauser et al. (1969)], Aspect et al. (1982) was using an experimental approach to test Bell Inequalities and to clearly show a significant gap between

After 40 years of development, many accurate experiments [Lindner et al. (2005); Zeh (1970); Zeilinger et al. (2005)] have been performed successfully worldwide using Laser, NMRI, large molecular, quantum coding and quantum communication approaches [Afshar et al. (2007); Barrow et al. (2004); Fox (2006); Merali (2007); Schleich et al. (2007)]. Following the application of advanced technologies and simulation methodologies, detailed single and multiple photon

However it does not matter how successful any single experiment or indeed many experiments might be, those results cannot simply replace the idea experiment of [Feynman et al. (1965,1989); Hawking & Mlodinow (2010); Penrose (2004)]. From a theoretical viewpoint, modern experiments involving Bell Inequations are excellent in illustrating the fundamental differences between a local realism and quantum reality. Since both theoretical and experimental activities focused on supporting or disproving Bell Inequalities cannot on their own provide a full explanation, further investigations are essential to provide a sound

In this chapter, a double path model has been established using the Mach-Zehnder interferometer. Different approaches to quantum measurements taken by Einstein, Stern-Gerlach, CHSH and Aspect are investigated to form quaternion structures. Using multiple-variable logic functions and variant principles, logic functions can be transferred into variant logic expressions as variant measures. Under such conditions, a variant simulation

foundation on which a full understanding of quantum issues can be constructed.

**1.4 From local interactive measurements to global matrix representations**

paradox under a local realism framework [Aspect et al. (1982); Bell (2004)].

**1.3 Modern experiments**

Grangier et al. (1986)].

**1.3.1 Bell approaches**

**1.3.2 Advanced experiments**

**1.3.3 Weakness**

Bell Inequalities and real quantum reality.

and representation model is proposed.

detection technologies have been further developed.

A Stern-Gerlach spin measurement device provides equivalent information for double path experiment [Jacques et al. (2008); Jammer (1974)]. This device divides composed signals into vertical ⊥ and horizontal � components, in *BP* part *ρ* → {*ρ*⊥, *ρ*�}, through controls and *IM* output *IM*(*ρ*⊥ *<sup>L</sup>* , *ρ* � *R*).

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

Meta Measurements MM

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

→ *ρ* →

<sup>375</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

→ {*PH*(*uβ*|*J*)} → → {*PH*(*vβ*|*J*)} →

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

(a) Architecture

(b) LIM Component

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

Distributions SD, Global Matrix Representations GMR respectively. The key part of the

The Meta Measurement (MM) component uses *<sup>N</sup>* bit 0-1 vector *<sup>X</sup>* and a given function *<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>*

MM transfers *N* bit 0-1 vector under *J*(*X*) to generate four Meta-measures, under a given Probability scheme, four probability measurements are generated to output as a quaternion

The Local Interactive Measurement (LIM) component is the key location for local interactions as shown in Figure 2(b) to transfer quaternion signal *ρ* under symmetry / anti-symmetry and synchronous / asynchronous conditions, in relation to four combination of time effects as (Left, Right, Double Particle, Double Wave) respectively. Two types of additive operations are

The Statistical Distribution (SD) component performs statistical activities on corresponding signals. It is necessary to exhaust all possible vectors of *X* with a total of 2*<sup>N</sup>* vectors. Under this construction, each sub-signal of {*u*, *v*} forms a special histogram with a one dimensional spectrum to indicate the distribution under function *J*. A total of eight histograms are

The Global Matrix Representation (GMR) component uses each statistical distribution of the

relevant probability histogram as an element of a matrix composed of a total of 22*<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 →

Local Interactive Measurements LIM

Global Matrix Representations GMR

→ *u* → *v*

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

→ {*M*(*uβ*)} → {*M*(*vβ*)}

2 ,

elements

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

> Statistical Distribution SD

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

→ *ρ*<sup>+</sup> →

identified. Each {*u*, *v*} signal is composed of four distinct signals.

*u* → *v* →

2 →

Interactive Measurement LIM Component

**3.1.2 Local Interactive Measurements**

**3.1.3 Statistical Distributions**

generated in the probability conditions.

**3.1.4 Global Matrix Representations**

system: the LIM component is shown in Fig 2(b).

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

**3.1.1 Meta Measurements**

signal *ρ*.

#### **2.2 Emission and absorption measurements of quantum interaction**

#### **2.2.1 Einstein measurements**

Einstein (1916) established the first model to describe atomic interaction with radiation. For two-state systems, Einstein's model is as follows. Let a system have two energy states: the ground state *E*<sup>1</sup> and the excited state *E*2. Let *N*<sup>1</sup> and *N*<sup>2</sup> be the average numbers of atoms in the ground and excited states respectively. The number of states are changed from emission state *<sup>E</sup>*<sup>2</sup> <sup>→</sup> *<sup>E</sup>*<sup>1</sup> with a rate *dN*<sup>21</sup> *dt* , and at any point in time, the number of ground states are determined by absorbed energies from *<sup>E</sup>*<sup>1</sup> <sup>→</sup> *<sup>E</sup>*<sup>2</sup> with a rate *dN*<sup>12</sup> *dt* respectively. For convenience of description, let *N*<sup>12</sup> be the number of atoms from *E*<sup>1</sup> to *E*<sup>2</sup> and *N*<sup>21</sup> be the numbers from *E*<sup>2</sup> to *E*1. In Einstein's model, a measurement quaternion is �*N*1, *N*2, *N*12, *N*21�.

#### **2.2.2 Spin measurements**

Uhlenback and Goudsmit proposed spin using devices devised by Stern-Gerlach [Cohn (1990); Jammer (1974)]. Spin can be represented by | ↑�, | ↓� in a two-state system. A quaternion ��↑ | ↑�,�↑ | ↓�,�↓ | ↑�,�↓ | ↓�� can be established for spin interactions.

CHSH proposed spin measures testing Bell Inequalities [Aspect (2002); Clauser et al. (1969)]. They applied ⊥→ + and �→ − to establish a measurement quaternion: �*N*++, *N*+−, *N*−+, *N*−−�. CHSH parameters are in the Stern-Gerlach scheme.

#### **2.2.3 Aspect's measurements**

Advanced experimental testing of Bell Inequalities for quantum measures were performed by [Aspect (2002); Aspect et al. (1982)]. In this set of experiments, active properties are measured via four measurements: transmission rate *Nt*, reflection rate *Nr*, correspondent rate *Nc* and also the total number *N<sup>ω</sup>* in *ω*-time period. This set of measurements is a quaternion �*Nt*, *Nr*, *Nc*, *Nω*�. Among these, *Nc* is a new data type not found in the Einstein and Stern-Gerlach schemes. As a matched pair of signals, this parameter indicates either single or double path issues. This parameter could be an extension of synchronous/asynchronous time-measurement.

#### **3. Variant simulation and representation system**

A comprehensive process of measurement from local interactions through to global matrix representations is described. It is hoped that this may offer a convenient path to assist theorists and experimenters seeking to devise experiments to further explore such natural mysteries through the application of sound principles of logic and measurement.

Using the variant principle described in the next subsections, 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 3.2-3.7 respectively.

#### **3.1 Simulation and representation model**

The full measurement and representation architecture as shown in Figure 2(a) is composed of four components: Meta Measurements MM, Local Interactive Measurements LIM, Statistical

$$\begin{array}{c} X \in B\_2^N \to \begin{array}{c} \text{Meta} \\ \text{Measurements} \end{array} \to \begin{array}{c} \text{[Local Interactive]} \\ \text{Meas currents} \end{array} \to \begin{array}{c} \text{[Meas Intermediate]} \\ \text{Meas currents} \end{array} \to u \\ u \to \begin{array}{c} \text{Statistical} \\ \text{Distribtion} \end{array} \to \begin{Bmatrix} P\_H(u\_\beta|f) \end{Bmatrix} \to \begin{Bmatrix} P\_H(u\_\beta|f) \end{Bmatrix} \to \begin{Bmatrix} \text{Semantics} \\ \text{GlobalMatrix} \end{Bmatrix} \\ \forall X \in B\_2^N \to \begin{array}{c} \text{SDM} \\ \text{SD} \end{array} \end{array}$$

$$\begin{array}{c} \text{(a) Archimedean} \end{array} \quad \forall f \in B\_2^{2^u} \to \begin{array}{c} \text{(a)Archimedean} \\ \text{GMR} \end{array} \to \begin{Bmatrix} M(u\_\beta) \end{Bmatrix}$$

$$\rho \to \begin{bmatrix} BP \\ \end{bmatrix} \to \rho\_- \to \begin{bmatrix} \text{SW} \\ \text{SW} \end{bmatrix} \to \rho\_+, (1-\rho\_-)/2 \to \begin{bmatrix} \text{M} \\ \text{IM} \end{bmatrix} \to v$$

#### (b) LIM Component

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

Distributions SD, Global Matrix Representations GMR respectively. The key part of the system: the LIM component is shown in Fig 2(b).

#### **3.1.1 Meta Measurements**

4 Measurement Systems

Einstein (1916) established the first model to describe atomic interaction with radiation. For two-state systems, Einstein's model is as follows. Let a system have two energy states: the ground state *E*<sup>1</sup> and the excited state *E*2. Let *N*<sup>1</sup> and *N*<sup>2</sup> be the average numbers of atoms in the ground and excited states respectively. The number of states are changed from emission

of description, let *N*<sup>12</sup> be the number of atoms from *E*<sup>1</sup> to *E*<sup>2</sup> and *N*<sup>21</sup> be the numbers from *E*<sup>2</sup>

Uhlenback and Goudsmit proposed spin using devices devised by Stern-Gerlach [Cohn (1990); Jammer (1974)]. Spin can be represented by | ↑�, | ↓� in a two-state system. A

CHSH proposed spin measures testing Bell Inequalities [Aspect (2002); Clauser et al. (1969)].

Advanced experimental testing of Bell Inequalities for quantum measures were performed by [Aspect (2002); Aspect et al. (1982)]. In this set of experiments, active properties are measured via four measurements: transmission rate *Nt*, reflection rate *Nr*, correspondent rate *Nc* and also the total number *N<sup>ω</sup>* in *ω*-time period. This set of measurements is a quaternion �*Nt*, *Nr*, *Nc*, *Nω*�. Among these, *Nc* is a new data type not found in the Einstein and Stern-Gerlach schemes. As a matched pair of signals, this parameter indicates either single or double path issues. This parameter could be an extension of synchronous/asynchronous

A comprehensive process of measurement from local interactions through to global matrix representations is described. It is hoped that this may offer a convenient path to assist theorists and experimenters seeking to devise experiments to further explore such natural mysteries

Using the variant principle described in the next subsections, 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

The full measurement and representation architecture as shown in Figure 2(a) is composed of four components: Meta Measurements MM, Local Interactive Measurements LIM, Statistical

quaternion ��↑ | ↑�,�↑ | ↓�,�↓ | ↑�,�↓ | ↓�� can be established for spin interactions.

*dt* , and at any point in time, the number of ground states are

*dt* respectively. For convenience

**2.2 Emission and absorption measurements of quantum interaction**

determined by absorbed energies from *<sup>E</sup>*<sup>1</sup> <sup>→</sup> *<sup>E</sup>*<sup>2</sup> with a rate *dN*<sup>12</sup>

to *E*1. In Einstein's model, a measurement quaternion is �*N*1, *N*2, *N*12, *N*21�.

They applied ⊥→ + and �→ − to establish a measurement quaternion: �*N*++, *N*+−, *N*−+, *N*−−�. CHSH parameters are in the Stern-Gerlach scheme.

**3. Variant simulation and representation system**

**3.1 Simulation and representation model**

through the application of sound principles of logic and measurement.

detailed principles and methods are described in Sections 3.2-3.7 respectively.

**2.2.1 Einstein measurements**

state *<sup>E</sup>*<sup>2</sup> <sup>→</sup> *<sup>E</sup>*<sup>1</sup> with a rate *dN*<sup>21</sup>

**2.2.2 Spin measurements**

**2.2.3 Aspect's measurements**

time-measurement.

The Meta Measurement (MM) component uses *<sup>N</sup>* bit 0-1 vector *<sup>X</sup>* and a given function *<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>* 2 , MM transfers *N* bit 0-1 vector under *J*(*X*) to generate four Meta-measures, under a given Probability scheme, four probability measurements are generated to output as a quaternion signal *ρ*.

#### **3.1.2 Local Interactive Measurements**

The Local Interactive Measurement (LIM) component is the key location for local interactions as shown in Figure 2(b) to transfer quaternion signal *ρ* under symmetry / anti-symmetry and synchronous / asynchronous conditions, in relation to four combination of time effects as (Left, Right, Double Particle, Double Wave) respectively. Two types of additive operations are identified. Each {*u*, *v*} signal is composed of four distinct signals.

#### **3.1.3 Statistical Distributions**

The Statistical Distribution (SD) component performs statistical activities on corresponding signals. It is necessary to exhaust all possible vectors of *X* with a total of 2*<sup>N</sup>* vectors. Under this construction, each sub-signal of {*u*, *v*} forms a special histogram with a one dimensional spectrum to indicate the distribution under function *J*. A total of eight histograms are generated in the probability conditions.

#### **3.1.4 Global Matrix Representations**

The Global Matrix Representation (GMR) component uses each statistical distribution of the relevant probability histogram as an element of a matrix composed of a total of 22*<sup>n</sup>* elements

*<sup>f</sup> <sup>f</sup>* <sup>∈</sup> <sup>3210</sup> *<sup>f</sup>*<sup>+</sup> <sup>∈</sup> 30 21 10 <sup>0</sup><sup>1</sup> *<sup>f</sup>*<sup>−</sup> <sup>∈</sup> *No*. *S*(*n*) 11 10 01 00 *S*0(*n*) 110 10<sup>1</sup> 010 001 *S*1(*n*) {∅} 0000 �∅| 1010 |3, 1� {0} 0001 �0| 1011 |3, 1� {∅} 0010 �∅| 1000 |3� {1, 0} 0011 �0| 1001 |3� {2} 0100 �2| 1110 |3, 1� {2, 0} 0101 �2, 0| 1111 |3, 1� {2, 1} 0110 �2| 1100 |3� {2, 1, 0} 0111 �2, 0| 1101 |3� {3} 1000 �∅| 0010 |1� {3, 0} 1001 �0| 0011 |1� {3, 1} 1010 �∅| 0000 |∅� {3, 1, 0} 1011 �0| 0001 |∅� {3, 2} 1100 �2| 0110 |1� {3, 2, 0} 1101 �2, 0| 0111 |1� {3, 2, 1} 1110 �2| 0100 |∅� {3, 2, 1, 0} 1111 �2, 0| 0101 |∅�

<sup>377</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

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

Under variant construction, *N* bits of 0-1 vector *X* under a function *J* produce four Meta

From a methodological viewpoint, this set of probability parameters belongs to *multivariate*

<sup>Δ</sup>*J*(*x*) = �Δ<sup>⊥</sup> *<sup>J</sup>*(*x*), <sup>Δ</sup><sup>+</sup> *<sup>J</sup>*(*x*), <sup>Δ</sup><sup>−</sup> *<sup>J</sup>*(*x*), <sup>Δ</sup>� *<sup>J</sup>*(*x*)� <sup>Δ</sup>*<sup>α</sup> <sup>J</sup>*(*x*) = 1, *<sup>J</sup>*(*x*) <sup>∈</sup> *<sup>J</sup>α*(*x*), *<sup>α</sup>* ∈ {⊥, <sup>+</sup>, <sup>−</sup>, �}

For any given *n*-variable state there is one position in Δ*J*(*x*) to be 1 and other 3 positions are

function *<sup>J</sup>*, *<sup>n</sup>* bit 0-1 output vector *<sup>Y</sup>*, *<sup>Y</sup>* = *<sup>J</sup>*(*X*) = �*J*+|*X*|*J*−�, *<sup>Y</sup>* = *YN*−1...*Yj*...*Y*0, 0 ≤ *<sup>j</sup> <*

Using four Meta measures, relevant probability measurements can be formulated. *<sup>ρ</sup>* = (*ρ*⊥, *<sup>ρ</sup>*+, *<sup>ρ</sup>*−, *<sup>ρ</sup>*�)=(*N*⊥/*N*, *<sup>N</sup>*+/*N*, *<sup>N</sup>*−/*N*, *<sup>N</sup>*�/*N*), 0 ≤ *<sup>ρ</sup>*⊥, *<sup>ρ</sup>*+, *<sup>ρ</sup>*−, *<sup>ρ</sup>*� ≤ 1.

<sup>Δ</sup> = �Δ⊥, <sup>Δ</sup>+, <sup>Δ</sup>−, <sup>Δ</sup>��

0, others

For any *<sup>N</sup>* bit 0-1 vector *<sup>X</sup>*, *<sup>X</sup>* <sup>=</sup> *XN*−1...*Xj*...*X*0, 0 <sup>≤</sup> *<sup>j</sup> <sup>&</sup>lt; <sup>N</sup>*, *Xj* <sup>∈</sup> *<sup>B</sup>*2, *<sup>X</sup>* <sup>∈</sup> *<sup>B</sup><sup>N</sup>*

Checking two functions *f* = 3 and *f* = 6 respectively. { *f* = 3 := �0|3�, *f*<sup>+</sup> = 11 := �0|∅�, *f*<sup>−</sup> = 2 := �∅|3�}; { *f* = 6 := �2|3�, *f*<sup>+</sup> = 14 := �2|∅�, *f*<sup>−</sup> = 2 := �∅|3�}.

(*<sup>X</sup>* : *<sup>J</sup>*(*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>*�

measures composed of a measure vector *N*

**3.3 Meta measures**

*probability measurements*.

0.

**3.3.1 Variant measure functions**

**3.3.2 Variant measures on vector**

Let Δ be the variant measure function

(3)

(4)

<sup>2</sup> under *n*-variable

for all possible functions {*J*}. In this configuration, C code schemes are applied to form a 22*n*−<sup>1</sup> <sup>×</sup> 22*<sup>n</sup>*−<sup>1</sup> matrix to show the selected distribution group.

Unlike the other coding schemes (SL, W, F, ...), only C code schemes provide a regular configuration to clearly differentiate the Left path as exhibiting horizontal actions and the Right path as exhibiting vertical actions . Such clearly polarized outcomes may have the potential to help in the understanding of interactive mechanism(s) between double path for particles and double path for waves properties.

#### **3.2 Variant principle**

The variant principle is based on *n*-variable logic functions [Zheng (2011); Zheng & Zheng (2010; 2011a;b); Zheng et al. (2011)].

#### **3.2.1 Two sets of states**

For any n-variables *<sup>x</sup>* = *xn*−1...*xi*...*x*0, 0 ≤ *<sup>i</sup> < <sup>n</sup>*, *xi* ∈ {0, 1} = *<sup>B</sup>*<sup>2</sup> let a position *<sup>j</sup>* be the selected bit 0 ≤ *j < n*, *xj* be the selected variable. Let output variable *y* and *n*-variable function *<sup>f</sup>* , *<sup>y</sup>* <sup>=</sup> *<sup>f</sup>*(*x*), *<sup>y</sup>* <sup>∈</sup> *<sup>B</sup>*2, *<sup>x</sup>* <sup>∈</sup> *<sup>B</sup><sup>n</sup>* <sup>2</sup> . For all states of *<sup>x</sup>*, a set *<sup>S</sup>*(*n*) composed of the 2*<sup>n</sup>* states can be divided into two sets: *S*0(*n*) and *S*1(*n*).

$$\begin{cases} S\_0(n) = \{ \mathbf{x} | \mathbf{x}\_j = \mathbf{0}, \forall \mathbf{x} \in B\_2^n \} \\ S\_1(n) = \{ \mathbf{x} | \mathbf{x}\_j = \mathbf{1}, \forall \mathbf{x} \in B\_2^n \} \\ S(n) = \{ S\_0(n), S\_1(n) \} \end{cases} \tag{1}$$

#### **3.2.2 Four meta functions**

For a given logic function *f* , input and output pair relationships define four meta logic functions { *<sup>f</sup>*⊥, *<sup>f</sup>*+, *<sup>f</sup>*−, *<sup>f</sup>*�}.

$$\begin{cases} f\_{\perp}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathcal{S}\_{0}(n), y = 0 \} \\ f\_{+}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathcal{S}\_{0}(n), y = 1 \} \\ f\_{-}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathcal{S}\_{1}(n), y = 0 \} \\ f\_{\top}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathcal{S}\_{1}(n), y = 1 \} \end{cases} \tag{2}$$

#### **3.2.3 Two polarized functions**

Considering two standard logic canonical expressions: AND-OR form is selected from { *<sup>f</sup>*+(*x*), *<sup>f</sup>*�(*x*)} as *<sup>y</sup>* = 1 items, and OR-AND form is selected from { *<sup>f</sup>*−(*x*), *<sup>f</sup>*⊥(*x*)} as *<sup>y</sup>* = <sup>0</sup> items. Considering { *<sup>f</sup>*�(*x*), *<sup>f</sup>*⊥(*x*)}, *xj* = *<sup>y</sup>* items, they are invariant themselves.

To select { *f*+(*x*), *f*−(*x*)}, *xj* �= *y* in forming variant logic expression. Let *f*(*x*) = �*f*+|*x*| *f*−� be a variant logic expression. Any logic function can be expressed as a variant logic form. In �*f*+|*x*| *f*−� structure, *f*<sup>+</sup> selected 1 items in *S*0(*n*) as same as the AND-OR standard expression, and *f*<sup>−</sup> selecting relevant parts the same as OR-AND expression 0 items in *S*1(*n*).

#### **3.2.4** *n* = 2 **representation**

For a convenient understanding of the variant representation, 2-variable logic structures are illustrated for its 16 functions as follows.


Checking two functions *f* = 3 and *f* = 6 respectively. { *f* = 3 := �0|3�, *f*<sup>+</sup> = 11 := �0|∅�, *f*<sup>−</sup> = 2 := �∅|3�}; { *f* = 6 := �2|3�, *f*<sup>+</sup> = 14 := �2|∅�, *f*<sup>−</sup> = 2 := �∅|3�}.

#### **3.3 Meta measures**

6 Measurement Systems

for all possible functions {*J*}. In this configuration, C code schemes are applied to form a

Unlike the other coding schemes (SL, W, F, ...), only C code schemes provide a regular configuration to clearly differentiate the Left path as exhibiting horizontal actions and the Right path as exhibiting vertical actions . Such clearly polarized outcomes may have the potential to help in the understanding of interactive mechanism(s) between double path for

The variant principle is based on *n*-variable logic functions [Zheng (2011); Zheng & Zheng

For any n-variables *<sup>x</sup>* = *xn*−1...*xi*...*x*0, 0 ≤ *<sup>i</sup> < <sup>n</sup>*, *xi* ∈ {0, 1} = *<sup>B</sup>*<sup>2</sup> let a position *<sup>j</sup>* be the selected bit 0 ≤ *j < n*, *xj* be the selected variable. Let output variable *y* and *n*-variable

*<sup>S</sup>*0(*n*) = {*x*|*xj* <sup>=</sup> 0, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>B</sup><sup>n</sup>*

*<sup>S</sup>*1(*n*) = {*x*|*xj* <sup>=</sup> 1, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>B</sup><sup>n</sup>*

For a given logic function *f* , input and output pair relationships define four meta logic

*<sup>f</sup>*⊥(*x*) = { *<sup>f</sup>*(*x*)|*<sup>x</sup>* ∈ *<sup>S</sup>*0(*n*), *<sup>y</sup>* = <sup>0</sup>} *f*+(*x*) = { *f*(*x*)|*x* ∈ *S*0(*n*), *y* = 1} *f*−(*x*) = { *f*(*x*)|*x* ∈ *S*1(*n*), *y* = 0} *<sup>f</sup>*�(*x*) = { *<sup>f</sup>*(*x*)|*<sup>x</sup>* ∈ *<sup>S</sup>*1(*n*), *<sup>y</sup>* = <sup>1</sup>}

Considering two standard logic canonical expressions: AND-OR form is selected from { *<sup>f</sup>*+(*x*), *<sup>f</sup>*�(*x*)} as *<sup>y</sup>* = 1 items, and OR-AND form is selected from { *<sup>f</sup>*−(*x*), *<sup>f</sup>*⊥(*x*)} as *<sup>y</sup>* = <sup>0</sup>

To select { *f*+(*x*), *f*−(*x*)}, *xj* �= *y* in forming variant logic expression. Let *f*(*x*) = �*f*+|*x*| *f*−� be a variant logic expression. Any logic function can be expressed as a variant logic form. In �*f*+|*x*| *f*−� structure, *f*<sup>+</sup> selected 1 items in *S*0(*n*) as same as the AND-OR standard expression,

For a convenient understanding of the variant representation, 2-variable logic structures are

items. Considering { *<sup>f</sup>*�(*x*), *<sup>f</sup>*⊥(*x*)}, *xj* = *<sup>y</sup>* items, they are invariant themselves.

and *f*<sup>−</sup> selecting relevant parts the same as OR-AND expression 0 items in *S*1(*n*).

*S*(*n*) = {*S*0(*n*), *S*1(*n*)}

<sup>2</sup> . For all states of *<sup>x</sup>*, a set *<sup>S</sup>*(*n*) composed of the 2*<sup>n</sup>* states

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2 }

2 }

matrix to show the selected distribution group.

particles and double path for waves properties.

(2010; 2011a;b); Zheng et al. (2011)].

function *<sup>f</sup>* , *<sup>y</sup>* <sup>=</sup> *<sup>f</sup>*(*x*), *<sup>y</sup>* <sup>∈</sup> *<sup>B</sup>*2, *<sup>x</sup>* <sup>∈</sup> *<sup>B</sup><sup>n</sup>*

can be divided into two sets: *S*0(*n*) and *S*1(*n*).

⎧ ⎨ ⎩

⎧ ⎪⎪⎨

⎪⎪⎩

22*n*−<sup>1</sup>

<sup>×</sup> 22*<sup>n</sup>*−<sup>1</sup>

**3.2 Variant principle**

**3.2.1 Two sets of states**

**3.2.2 Four meta functions**

functions { *<sup>f</sup>*⊥, *<sup>f</sup>*+, *<sup>f</sup>*−, *<sup>f</sup>*�}.

**3.2.3 Two polarized functions**

**3.2.4** *n* = 2 **representation**

illustrated for its 16 functions as follows.

Under variant construction, *N* bits of 0-1 vector *X* under a function *J* produce four Meta measures composed of a measure vector *N*

$$(X:J(X)) \to (N\_{\perp \prime}N\_{+\prime}N\_{-\prime}N\_{\top}), \\ N = N\_{\perp} + N\_{+} + N\_{-} + N\_{\top}$$

Using four Meta measures, relevant probability measurements can be formulated. *<sup>ρ</sup>* = (*ρ*⊥, *<sup>ρ</sup>*+, *<sup>ρ</sup>*−, *<sup>ρ</sup>*�)=(*N*⊥/*N*, *<sup>N</sup>*+/*N*, *<sup>N</sup>*−/*N*, *<sup>N</sup>*�/*N*), 0 ≤ *<sup>ρ</sup>*⊥, *<sup>ρ</sup>*+, *<sup>ρ</sup>*−, *<sup>ρ</sup>*� ≤ 1.

From a methodological viewpoint, this set of probability parameters belongs to *multivariate probability measurements*.

#### **3.3.1 Variant measure functions**

Let Δ be the variant measure function

$$\begin{array}{l} \Delta = \langle \Delta\_{\perp}, \Delta\_{+}, \Delta\_{-}, \Delta\_{\top} \rangle \\ \Delta f(\mathbf{x}) = \langle \Delta\_{\perp} f(\mathbf{x}), \Delta\_{+} f(\mathbf{x}), \Delta\_{-} f(\mathbf{x}), \Delta\_{\top} f(\mathbf{x}) \rangle \\ \Delta\_{\mathbf{a}} f(\mathbf{x}) = \begin{cases} 1, f(\mathbf{x}) \in f\_{\mathbf{a}}(\mathbf{x}), \mathbf{a} \in \{\perp, +, -, \top\} \\ 0, \text{others} \end{cases} \end{array} \tag{4}$$

For any given *n*-variable state there is one position in Δ*J*(*x*) to be 1 and other 3 positions are 0.

#### **3.3.2 Variant measures on vector**

For any *<sup>N</sup>* bit 0-1 vector *<sup>X</sup>*, *<sup>X</sup>* <sup>=</sup> *XN*−1...*Xj*...*X*0, 0 <sup>≤</sup> *<sup>j</sup> <sup>&</sup>lt; <sup>N</sup>*, *Xj* <sup>∈</sup> *<sup>B</sup>*2, *<sup>X</sup>* <sup>∈</sup> *<sup>B</sup><sup>N</sup>* <sup>2</sup> under *n*-variable function *<sup>J</sup>*, *<sup>n</sup>* bit 0-1 output vector *<sup>Y</sup>*, *<sup>Y</sup>* = *<sup>J</sup>*(*X*) = �*J*+|*X*|*J*−�, *<sup>Y</sup>* = *YN*−1...*Yj*...*Y*0, 0 ≤ *<sup>j</sup> <*

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⎧

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

*u*+ = *ρ*+ *u*<sup>−</sup> = *ρ*<sup>−</sup> *u*<sup>0</sup> = *u*<sup>+</sup> ⊕ *u*<sup>−</sup> *u*<sup>1</sup> = *u*<sup>+</sup> + *u*<sup>−</sup> *<sup>v</sup>*<sup>+</sup> <sup>=</sup> <sup>1</sup>+*ρ*<sup>+</sup> 2 *<sup>v</sup>*<sup>−</sup> <sup>=</sup> <sup>1</sup>−*ρ*<sup>−</sup> 2 *v*<sup>0</sup> = *v*<sup>+</sup> ⊕ *v*<sup>−</sup> *v*<sup>1</sup> = *v*<sup>+</sup> + *v*<sup>−</sup> − 0.5

<sup>379</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

where 0 ≤ *uβ*, *v<sup>β</sup>* ≤ 1, *β* ∈ {+, −, 0, 1}, ⊕ : Asynchronous addition, + : Synchronous addition.

The SD component provides a statistical means to accumulate all possible vectors of *N* bits for a selected signal and generate a histogram. Eight signals correspond to eight histograms respectively. Among these, four histograms exhibit properties of symmetry and another four

For a function *J*, all measurement signals are collected and the relevant histogram represents

Using *u* and *v* signals, each *u<sup>β</sup>* or *v<sup>β</sup>* determines a fixed position in the relevant histogram to make vector *X* on a position. After completing 2*<sup>N</sup>* data sequences, eight

Let |*H*(..)| denote the total number in the histogram *H*(..), a normalized Probability histogram

Distributions are dependant on the data set as a whole and are not sensitive to varying under special sequences. Under this condition, when the data set has been exhaustively listed, then

The eight histogram distributions provide invariant spectrum to represent properties among



<sup>2</sup> areas respectively.

*PH*(*uβ*|*J*) = *<sup>H</sup>*(*uβ*|*J*)

*PH*(*vβ*|*J*) = *<sup>H</sup>*(*vβ*|*J*)

<sup>2</sup> *<sup>H</sup>*(*uβ*|*J*(*X*))

<sup>2</sup> *<sup>H</sup>*(*vβ*|*J*(*X*)), *<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>*

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

2

symmetry/anti-symmetry histograms of {*H*(*uβ*|*J*)}, {*H*(*vβ*|*J*)} are generated.

� *<sup>H</sup>*(*uβ*|*J*) = <sup>∑</sup>∀*X*∈*B<sup>N</sup>*

⎧ ⎨ ⎩

the same distributions are always linked to the given signal set.

*<sup>H</sup>*(*vβ*|*J*) = <sup>∑</sup>∀*X*∈*B<sup>N</sup>*

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⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

**3.6 Statistical distributions**

**3.6.1 Statistical histograms**

a complete statistical distribution.

For a function *J*, *β* ∈ {+, −, 0, 1}

**3.6.2 Probability histograms**

(*PH*(..)) can be expressed as

Here, all histograms are restricted in [0, 1]

different interactive conditions.

histograms exhibit properties of anti-symmetry.

*<sup>N</sup>*,*Yj* <sup>∈</sup> *<sup>B</sup>*2,*<sup>Y</sup>* <sup>∈</sup> *<sup>B</sup><sup>N</sup>* <sup>2</sup> . For the *<sup>j</sup>*-th position *<sup>x</sup><sup>j</sup>* = [...*Xj*...] <sup>∈</sup> *<sup>B</sup><sup>n</sup>* <sup>2</sup> to form *Yj* = *<sup>J</sup>*(*x<sup>j</sup>* ) = �*J*+|*x<sup>j</sup>* |*J*−�. Let *N* bit positions be cyclic linked. Variant measures of *J*(*X*) can be decomposed

$$\Delta(X:Y) = \Delta f(X) = \sum\_{j=0}^{N-1} \Delta f(x^j) = \langle N\_{\perp \prime} N\_{+\prime} N\_{-\prime} N\_{\top} \rangle \tag{5}$$

as a quaternion �*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>*�.

#### **3.3.3 Example**

\*\*E.g.\*\*  $N = 12$ , given  $f\_\* Y = f(X)$ .

$$X \quad = 1 \ 0 \ 1 \ 1 \ 1 \ 1 \ 0 \ 1 \ 1 \ 1 \ 1 \ 1 \ 0 \ 0 \ 0 \ 1$$

$$Y \quad = 0 \ 0 \ 0 \ 1 \ 0 \ 1 \ 0 \ 1 \ 0 \ 1 \ 0 \ 1 \ 1 \ 0 \ 0$$

$$\Delta(X:Y) = - \ \bot \ \top - \top \ \bot \ \top - \top + \bot -$$

$$\Delta f(X) = \langle N\_{\perp \prime} N\_{+\prime} N\_{-\prime} N\_{\top} \rangle = \langle 3, 1, 4, 4 \rangle, N = 12.$$

Input and output pairs are 0-1 variables for only four combinations. For any given function *J*, the quantitative relationship of {⊥, +, −, �} is directly derived from the input/output sequences. Four meta measures are determined.

#### **3.4 Four meta measurements**

Using variant quaternion, local measurements of probability signals are calculated as four meta measurements by following the given equations. For any *N* bit 0-1 vector *X*, function *J*, under <sup>Δ</sup> measurement: <sup>Δ</sup>*J*(*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>*�

Signal *ρ* is defined by

$$\begin{cases} \rho = \frac{\Delta I(X)}{N} = (\rho\_{\perp \prime} \rho\_{+\prime} \rho\_{-\prime} \rho\_{\top}) \\\\ \rho\_{\alpha} = \frac{N\_{\theta}}{N}, a \in \{\perp, +, -, \top\} \end{cases} \tag{6}$$

The four meta measurements are core components in the *multivariate probability framework*.

#### **3.5 Local Interactive Measurements**

Local Interactive Measurements (LIM) are divided into three stages: BP, SW and SM respectively. The BP stage selects {*ρ*−, *ρ*+} as sub-signals. The SW component extends two signals into four signals with different symmetric properties; The SM component merges different signals to form two sets of eight signals.

Using {*ρ*+, *ρ*−}, a pair of signals {*u*, *v*} are formulated:

$$\begin{cases} u = (u\_{+\prime}, u\_{-\prime}, u\_0, u\_1) = \{u\_{\beta}\} \\ v = (v\_{+\prime}v\_{-\prime}, v\_{0\prime}v\_1) = \{v\_{\beta}\} \\ \mathcal{B} \in \{+, -, 0, 1\} \end{cases} \tag{7}$$

$$\begin{cases} u\_+ = \rho\_+ \\ u\_- = \rho\_- \\ u\_0 = u\_+ \oplus u\_- \\ u\_1 = u\_+ + u\_- \\ v\_+ = \frac{1+\rho\_+}{2} \\ v\_- = \frac{1-\rho\_-}{2} \\ v\_0 = v\_+ \oplus v\_- \\ v\_1 = v\_+ + v\_- - 0.5 \end{cases} \tag{8}$$

where 0 ≤ *uβ*, *v<sup>β</sup>* ≤ 1, *β* ∈ {+, −, 0, 1}, ⊕ : Asynchronous addition, + : Synchronous addition.

#### **3.6 Statistical distributions**

8 Measurement Systems

Δ*J*(*x<sup>j</sup>*

*X* = 101110111001 *Y* = 001010101100

Δ(*X* : *Y*) = −⊥�−�⊥�−� + ⊥ −

Input and output pairs are 0-1 variables for only four combinations. For any given function *J*, the quantitative relationship of {⊥, +, −, �} is directly derived from the input/output

Using variant quaternion, local measurements of probability signals are calculated as four meta measurements by following the given equations. For any *N* bit 0-1 vector *X*, function *J*,

*<sup>N</sup>* = (*ρ*⊥, *<sup>ρ</sup>*+, *<sup>ρ</sup>*−, *<sup>ρ</sup>*�)

*<sup>N</sup>* , *α* ∈ {⊥, +, −, �}

The four meta measurements are core components in the *multivariate probability framework*.

Local Interactive Measurements (LIM) are divided into three stages: BP, SW and SM respectively. The BP stage selects {*ρ*−, *ρ*+} as sub-signals. The SW component extends two signals into four signals with different symmetric properties; The SM component merges

> *u* = (*u*+, *u*−, *u*0, *u*1) = {*uβ*} *v* = (*v*+, *v*−, *v*0, *v*1) = {*vβ*} *β* ∈ {+, −, 0, 1}

under <sup>Δ</sup> measurement: <sup>Δ</sup>*J*(*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>*�

*ρ* = <sup>Δ</sup>*J*(*X*)

*ρα* = *<sup>N</sup><sup>α</sup>*

⎧ ⎨ ⎩ <sup>2</sup> to form *Yj* = *<sup>J</sup>*(*x<sup>j</sup>*

) = �*N*⊥, *<sup>N</sup>*+, *<sup>N</sup>*−, *<sup>N</sup>*�� (5)

) = �*J*+|*x<sup>j</sup>*


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(7)

<sup>2</sup> . For the *<sup>j</sup>*-th position *<sup>x</sup><sup>j</sup>* = [...*Xj*...] <sup>∈</sup> *<sup>B</sup><sup>n</sup>*

Δ(*X* : *Y*) = Δ*J*(*X*) =

as a quaternion �*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>*�.

<sup>Δ</sup>*J*(*X*) = �*N*⊥, *<sup>N</sup>*+, *<sup>N</sup>*−, *<sup>N</sup>*�� = �3, 1, 4, 4�, *<sup>N</sup>* = 12.

sequences. Four meta measures are determined.

Let *N* bit positions be cyclic linked. Variant measures of *J*(*X*) can be decomposed

*N*−1 ∑ *j*=0

*<sup>N</sup>*,*Yj* <sup>∈</sup> *<sup>B</sup>*2,*<sup>Y</sup>* <sup>∈</sup> *<sup>B</sup><sup>N</sup>*

**3.3.3 Example**

E.g. *N* = 12, given *J*,*Y* = *J*(*X*).

**3.4 Four meta measurements**

**3.5 Local Interactive Measurements**

different signals to form two sets of eight signals.

Using {*ρ*+, *ρ*−}, a pair of signals {*u*, *v*} are formulated:

⎧ ⎪⎪⎨

⎪⎪⎩

Signal *ρ* is defined by

The SD component provides a statistical means to accumulate all possible vectors of *N* bits for a selected signal and generate a histogram. Eight signals correspond to eight histograms respectively. Among these, four histograms exhibit properties of symmetry and another four histograms exhibit properties of anti-symmetry.

#### **3.6.1 Statistical histograms**

For a function *J*, all measurement signals are collected and the relevant histogram represents a complete statistical distribution.

Using *u* and *v* signals, each *u<sup>β</sup>* or *v<sup>β</sup>* determines a fixed position in the relevant histogram to make vector *X* on a position. After completing 2*<sup>N</sup>* data sequences, eight symmetry/anti-symmetry histograms of {*H*(*uβ*|*J*)}, {*H*(*vβ*|*J*)} are generated.

For a function *J*, *β* ∈ {+, −, 0, 1}

$$\begin{cases} H(u\_{\beta}|J) = \sum\_{\forall X \in B\_2^N} H(u\_{\beta}|J(X)) \\ H(v\_{\beta}|J) = \sum\_{\forall X \in B\_2^N} H(v\_{\beta}|J(X))\_{\prime} \int \in B\_2^{2^{\prime}} \end{cases} \tag{9}$$

#### **3.6.2 Probability histograms**

Let |*H*(..)| denote the total number in the histogram *H*(..), a normalized Probability histogram (*PH*(..)) can be expressed as

$$\begin{cases} P\_H(u\_\beta|J) = \frac{H(u\_\beta|I)}{|H(u\_\beta|I)|} \\ P\_H(v\_\beta|J) = \frac{H(v\_\beta|I)}{|H(v\_\beta|I)|}, J \in \mathcal{B}\_2^{2^n} \end{cases} \tag{10}$$

Here, all histograms are restricted in [0, 1] <sup>2</sup> areas respectively.

Distributions are dependant on the data set as a whole and are not sensitive to varying under special sequences. Under this condition, when the data set has been exhaustively listed, then the same distributions are always linked to the given signal set.

The eight histogram distributions provide invariant spectrum to represent properties among different interactive conditions.

*PH*(*u*+|*J*) *PH*(*u*−|*J*) (a) Left (b) Right *PH*(*u*0|*J*) *PH*(*u*1|*J*) (c) D-P (d) D-W *PH*(*v*+|*J*) *PH*(*v*−|*J*) (e) Left (f) Right *PH*(*v*0|*J*) *PH*(*v*1|*J*) (g) D-P (h) D-W (I) Representative patterns of Histograms for function *J* (a-d) symmetric cases; (e-h) antisymmetric cases

<sup>381</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

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

(a) Left (b) Right

(c) D-P (d) D-W

(e) Left (f) Right

(g) D-P (h) D-W (II) *N* = {12}, *J* = 3 Two groups of results in eight histograms

#### **3.7 Global Matrix Representations**

After local interactive measurements and statistical process are undertaken for a given function *J*, eight histograms are generated. The Global Matrix Representation GMR component performs its operations into two stages. In the first stage, exhausting all possible functions for <sup>∀</sup>*<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>* <sup>2</sup> to generate eight sets, each set contains 22*<sup>n</sup>* elements and each element is a histogram. In the second stage, arranging all 22*<sup>n</sup>* elements generated as a matrix by C code scheme. Here, we can see Left and Right path reactions polarized into Horizontal and Vertical relationships respectively.

#### **3.7.1 Matrix and Its elements**

For a given C scheme, let *<sup>C</sup>*(*J*) = �*J*1|*J*0�, each element

$$\begin{cases} \mathcal{M}\_{\langle I^1 | J^0 \rangle} (\boldsymbol{u}\_{\boldsymbol{\beta}} | \boldsymbol{J}) = P\_H (\boldsymbol{u}\_{\boldsymbol{\beta}} | \boldsymbol{J}) \\ \mathcal{M}\_{\langle I^1 | J^0 \rangle} (\boldsymbol{v}\_{\boldsymbol{\beta}} | \boldsymbol{J}) = P\_H (\boldsymbol{v}\_{\boldsymbol{\beta}} | \boldsymbol{J}) \\ \boldsymbol{J} \in \mathcal{B}\_2^{2^n}; \; \boldsymbol{J}^1, \boldsymbol{J}^0 \in \mathcal{B}\_2^{2^{n-1}} \end{cases} \tag{11}$$

#### **3.7.2 Representation patterns of matrices**

For example, using *n* = 2, *P* = (3102), Δ = (1111) conditions, a C code case contains sixteen histograms arranged as a 4 × 4 matrix.

$$
\begin{array}{|c|c|c|c|}
\hline
0 & 4 & 1 & 5 \\
\hline
2 & 6 & 3 & 7 \\
\hline
8 & 12 & 9 & 13 \\
\hline
10 & 14 & 11 & 15 \\
\hline
\end{array}
\tag{12}
$$

All matrices in this chapter use this configuration for the matrix pattern to represent their elements.
