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

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 of logic and measurement.

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 and group distributions in multivariate probability environments.

#### **4.1 Conditional simulation and representation model**

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

#### **4.1.1 Conditional Meta Measurements**

The Conditional Meta Measurement (CMM) component uses *N* bit 0-1 vector *X* and a given function *<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>* <sup>2</sup> , CMM transfers *N* bit 0-1 vector under *J*(*X*) to generate four Meta-measures,

**4.2.2 Four meta functions**

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

**4.2.3 Two polarized functions**

⎧ ⎪⎪⎨

⎪⎪⎩

**4.3 Meta measures and conditional probability measurements**

measures composed of a measure vector *N*

**4.3.1 Variant measure functions**

**4.3.2 Variant measures on vector**

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

0.

Let Δ be the variant measure function

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

<sup>347</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

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

*<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: the AND-OR form is selected from { *<sup>f</sup>*+(*x*), *<sup>f</sup>*�(*x*)} as *<sup>y</sup>* = 1 items, and the OR-AND form is selected from { *<sup>f</sup>*−(*x*), *<sup>f</sup>*⊥(*x*)} as

To select { *f*+(*x*), *f*−(*x*)}, *xj* �= *y* in forming a 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 the 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*).

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

<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> <*

(*<sup>X</sup>* : *<sup>J</sup>*(*X*)) → (*N*⊥, *<sup>N</sup>*+, *<sup>N</sup>*−, *<sup>N</sup>*�), *<sup>N</sup>*<sup>0</sup> = *<sup>N</sup>*<sup>⊥</sup> + *<sup>N</sup>*+, *<sup>N</sup>*<sup>1</sup> = *<sup>N</sup>*<sup>−</sup> + *<sup>N</sup>*�, *<sup>N</sup>* = *<sup>N</sup>*<sup>0</sup> + *<sup>N</sup>*<sup>1</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*0, *<sup>N</sup>*+/*N*0, *<sup>N</sup>*−/*N*1, *<sup>N</sup>*�/*N*1), 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>*

<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>y</sup>* = 0 items. Considering { *<sup>f</sup>*�(*x*), *<sup>f</sup>*⊥(*x*)}, *xj* = *<sup>y</sup>* items, they are themselves invariant.

(6)

(7)


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

) = �*J*+|*x<sup>j</sup>*

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

under a given probability scheme, four conditional probability measurements are generated and output as a quaternion signal *ρ*˜.

#### **4.1.2 Conditional Interactive Measurements**

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

#### **4.1.3 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.

#### **4.1.4 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 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.

#### **4.2 Variant principle**

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

#### **4.2.1 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*).

$$\begin{cases} S\_0(n) = \{ \mathbf{x} | \mathbf{x}\_j = \mathbf{0}, \forall \mathbf{x} \in B\_2^n \} \\ S\_1(n) = \{ \mathbf{x} | \mathbf{x}\_j = \mathbf{1}, \forall \mathbf{x} \in B\_2^n \} \\ S(n) = \{ S\_0(n), S\_1(n) \} \end{cases} \tag{5}$$

#### **4.2.2 Four meta functions**

8 Measurement Systems

under a given probability scheme, four conditional probability measurements are generated

The Conditional Interactive Measurement (CIM) component is the key location for conditional interactions as shown in Figure 2(b) to transfer a quaternion signal *ρ*˜ under symmetry / anti-symmetry and synchronous / asynchronous conditions, under four combinations of time effects namely (Left, Right, Double Particle, Double Wave). Two types of additive operations

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

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>*

*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

2 }

2 }

elements

(5)

relevant probability histogram as an element of a matrix composed of a total of 22*<sup>n</sup>*

matrix to show the selected distribution group.

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

and output as a quaternion signal *ρ*˜.

**4.1.3 Statistical Distributions**

generated in the probability conditions.

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*).

⎧ ⎨ ⎩

**4.1.4 Global Matrix Representations**

22*n*−<sup>1</sup>

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

**4.2 Variant principle**

**4.2.1 Two sets of states**

**4.1.2 Conditional Interactive Measurements**

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>*�}.

$$\begin{cases} f\_{\perp}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathcal{S}\_{0}(n), y = 0 \} \\ f\_{+}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathcal{S}\_{0}(n), y = 1 \} \\ f\_{-}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathcal{S}\_{1}(n), y = 0 \} \\ f\_{\top}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathcal{S}\_{1}(n), y = 1 \} \end{cases} \tag{6}$$

#### **4.2.3 Two polarized functions**

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

To select { *f*+(*x*), *f*−(*x*)}, *xj* �= *y* in forming a 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 the 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*).

#### **4.3 Meta measures and conditional probability measurements**

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

$$(X:J(X)) \rightarrow (N\_{\perp'}N\_{+'}N\_{-'}N\_{\top})\_{\prime}N\_0 = N\_{\perp} + N\_{+'}N\_1 = N\_{-} + N\_{\top'}N = N\_0 + N\_1$$

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*0, *<sup>N</sup>*+/*N*0, *<sup>N</sup>*−/*N*1, *<sup>N</sup>*�/*N*1), 0 ≤ *<sup>ρ</sup>*˜⊥, *<sup>ρ</sup>*˜+, *<sup>ρ</sup>*˜−, *<sup>ρ</sup>*˜� ≤ 1.

#### **4.3.1 Variant measure functions**

Let Δ be the variant measure function

$$\begin{array}{l} \Delta = \langle \Delta\_{\perp}, \Delta\_{+}, \Delta\_{-}, \Delta\_{\top} \rangle \\ \Delta f(\mathbf{x}) = \langle \Delta\_{\perp} f(\mathbf{x}), \Delta\_{+} f(\mathbf{x}), \Delta\_{-} f(\mathbf{x}), \Delta\_{\top} f(\mathbf{x}) \rangle \\ \Delta\_{\mathbf{a}} f(\mathbf{x}) = \begin{cases} 1, f(\mathbf{x}) \in f\_{\mathbf{a}}(\mathbf{x}), \mathbf{a} \in \{\perp, +, -, \top\} \\ 0, \text{others} \end{cases} \end{array} \tag{7}$$

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

#### **4.3.2 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> < <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*−�.

⎧

*u*˜+ = *ρ*˜+ *u*˜<sup>−</sup> = *ρ*˜<sup>−</sup> *u*˜0 = *u*˜<sup>+</sup> ⊕ *u*˜<sup>−</sup> *u*˜1 = (*u*˜<sup>+</sup> + *u*˜−)/2

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

<sup>349</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

*<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*˜0 = *v*˜<sup>+</sup> ⊕ *v*˜<sup>−</sup> *v*˜1 = *v*˜<sup>+</sup> + *v*˜<sup>−</sup> − 0.5

where 0 ≤ *u*˜*β*, *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 the other 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



*PH*(*u*˜*β*|*J*) = *<sup>H</sup>*(*u*˜*β*|*J*)

*PH*(*v*˜*β*|*J*) = *<sup>H</sup>*(*v*˜*β*|*J*)

<sup>2</sup> areas.

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 spectra to represent properties among

<sup>2</sup> *<sup>H</sup>*(*u*˜*β*|*J*(*X*))

<sup>2</sup> *<sup>H</sup>*(*v*˜*β*|*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*(*u*˜*β*|*J*)}, {*H*(*v*˜*β*|*J*)} are generated.

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

⎧ ⎨ ⎩

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

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

(11)

(12)

(13)

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

**4.6 Statistical distributions**

**4.6.1 Statistical histograms**

a complete statistical distribution.

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

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

**4.6.2 Normalized probability histograms**

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

different interactive conditions.

histograms exhibit properties of anti-symmetry.

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

$$\Delta(X:Y) = \Delta f(X) = \sum\_{j=0}^{N-1} \Delta f(x^j) = \langle N\_{\perp \prime} N\_{+\prime} N\_{-\prime} N\_{\top} \rangle \tag{8}$$

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>*�.

#### **4.3.3 Example**

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


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.

#### **4.4 Four conditional meta measurements**

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

Signal *ρ*˜ is defined by

$$\begin{cases} \vec{\rho} = (\vec{\rho}\_{\perp'} \vec{\rho}\_{+} , \vec{\rho}\_{-} \vec{\rho}\_{-} \vec{\rho}\_{\top}) \\ \vec{\rho}\_{\perp} = \frac{N\_{\perp}}{N\_{0}} \\ \vec{\rho}\_{+} = \frac{N\_{\perp}}{N\_{0}} \\ \vec{\rho}\_{-} = \frac{N\_{\perp}}{N\_{1}} \end{cases} \tag{9}$$

#### **4.5 Conditional Interactive Measurements**

Conditional Interactive Measurements (CIM) 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:

$$\begin{cases} \mathfrak{u} = (\mathfrak{u}\_{+}, \mathfrak{u}\_{-}, \mathfrak{u}\_{0}, \mathfrak{v}\_{1}) = \{u\_{\beta}\} \\ \mathfrak{v} = (\mathfrak{v}\_{+}, \mathfrak{v}\_{-}, \mathfrak{v}\_{0}, \mathfrak{v}\_{1}) = \{v\_{\beta}\} \\ \mathcal{B} \in \{+, -, 0, 1\} \end{cases} \tag{10}$$

$$\begin{cases} \begin{aligned} \tilde{u}\_{+} &= \tilde{\rho}\_{+} \\ \tilde{u}\_{-} &= \tilde{\rho}\_{-} \\ \tilde{u}\_{0} &= \tilde{u}\_{+} \oplus \tilde{u}\_{-} \\ \tilde{u}\_{1} &= (\tilde{u}\_{+} + \tilde{u}\_{-})/2 \\ \tilde{v}\_{+} &= \frac{1 + \tilde{\rho}\_{+}}{2} \\ \tilde{v}\_{-} &= \frac{1 - \tilde{\rho}\_{-}}{2} \\ \tilde{v}\_{0} &= \tilde{v}\_{+} \oplus \tilde{v}\_{-} \\ \tilde{v}\_{1} &= \tilde{v}\_{+} + \tilde{v}\_{-} - 0.5 \end{aligned} \tag{11}$$

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

#### **4.6 Statistical distributions**

10 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, conditional measurements of probability signals are calculated as four meta conditional measurements by following the given equations. For any *N* bit 0-1 vector *<sup>X</sup>*, function *<sup>J</sup>*, under <sup>Δ</sup> measurement: <sup>Δ</sup>*J*(*X*) = �*N*⊥, *<sup>N</sup>*+, *<sup>N</sup>*−, *<sup>N</sup>*��, *<sup>N</sup>*<sup>0</sup> = *<sup>N</sup>*<sup>⊥</sup> + *<sup>N</sup>*+,

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

Conditional Interactive Measurements (CIM) 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}

) = �*N*⊥, *<sup>N</sup>*+, *<sup>N</sup>*−, *<sup>N</sup>*�� (8)

(9)

(10)

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

*N*−1 ∑ *j*=0

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.

*<sup>N</sup>*<sup>1</sup> = *<sup>N</sup>*<sup>−</sup> + *<sup>N</sup>*�, *<sup>N</sup>* = *<sup>N</sup>*<sup>0</sup> + *<sup>N</sup>*<sup>1</sup> = *<sup>N</sup>*<sup>⊥</sup> + *<sup>N</sup>*<sup>+</sup> + *<sup>N</sup>*<sup>−</sup> + *<sup>N</sup>*�.

⎧ ⎪⎪⎪⎪⎪⎨

*<sup>ρ</sup>*˜<sup>⊥</sup> <sup>=</sup> *<sup>N</sup>*<sup>⊥</sup> *N*<sup>0</sup> *<sup>ρ</sup>*˜<sup>+</sup> = *<sup>N</sup>*<sup>+</sup> *N*<sup>0</sup> *<sup>ρ</sup>*˜<sup>−</sup> <sup>=</sup> *<sup>N</sup>*<sup>−</sup> *N*<sup>1</sup> *<sup>ρ</sup>*˜� <sup>=</sup> *<sup>N</sup>*� *N*<sup>1</sup>

⎪⎪⎪⎪⎪⎩

**4.4 Four conditional meta measurements**

**4.5 Conditional 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

**4.3.3 Example**

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

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 the other four histograms exhibit properties of anti-symmetry.

#### **4.6.1 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*(*u*˜*β*|*J*)}, {*H*(*v*˜*β*|*J*)} are generated.

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

$$\begin{cases} H(\mathfrak{u}\_{\beta}|I) = \sum\_{\forall X \in B\_2^N} H(\mathfrak{u}\_{\beta}|I(X)) \\ H(\mathfrak{v}\_{\beta}|I) = \sum\_{\forall X \in B\_2^N} H(\mathfrak{v}\_{\beta}|I(X)) \,, I \in B\_2^{2^u} \end{cases} \tag{12}$$

#### **4.6.2 Normalized probability histograms**

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

$$\begin{cases} P\_H(\tilde{u}\_\beta | J) = \frac{H(\vec{u}\_\beta | I)}{|H(\vec{u}\_\beta | I)|} \\ P\_H(\tilde{v}\_\beta | J) = \frac{H(\vec{v}\_\beta | J)}{|H(\vec{v}\_\beta | I)|}, J \in \mathcal{B}\_2^{2^n} \end{cases} \tag{13}$$

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

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 spectra 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>351</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Intrinsic Quantum Waves 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

#### **4.7 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.

#### **4.7.1 Matrix and its elements**

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

$$\begin{cases} M\_{\langle f^1 | f^0 \rangle} (\vec{u}\_{\beta} | I) = P\_H(\vec{u}\_{\beta} | I) \\ M\_{\langle f^1 | f^0 \rangle} (\vec{v}\_{\beta} | I) = P\_H(\vec{v}\_{\beta} | I) \\ J \in B\_2^{2^n}; J^1, J^0 \in B\_2^{2^{n-1}} \end{cases} \tag{14}$$

#### **4.7.2 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.

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

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