Keywords

1 Introduction

Feynman explored quantum measurement puzzles deeply [1, 2] and emphasized: “The entire mystery of quantum mechanics is in the double-slit experiment.” This experiment directly illustrates both classical and quantum interactive results. Under single and double slit conditions, dual visual distributions are shown in particle and wave statistical distributions linked to von Neumann’s measure theory [3].

From the 1970s, piloted by CHSH [4], Aspect used experiments to test Bell inequalities [5,6,7]. After 40 years of development, many accurate experiments [8,9,10] have been performed successfully worldwide using Laser, NMRI, large molecular, quantum coding, and quantum communication approaches [5,6,7,8, 11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].

In this chapter, a double path model is established using the Mach–Zehnder interferometer. Different approaches of quantum measures: Einstein, CHSH, and Aspect are investigated by quaternion structures. Under multiple-variable logic functions and variant principle, logic functions can be transferred into variant logic expression as variant measures. Under such conditions, a variant simulation model is proposed. A given logic function \( f \) can be represented as two meta-logic functions \( f_{ + } \) and \( f_{ - } \) to simulate single and double path conditions. \( N \) bits of input vectors are exhausted by \( 2^{N} \) states for measured data, recursive data are organized into eight histograms. Results are determined by symmetry/anti-symmetry properties evident in these histograms. Both results are obtained consistently from this model on synchronous/asynchronous conditions. Based on this set of simulation results, synchronous condition shows significant relationship linked to interference properties.

2 Double Path Model and Their Measures

2.1 Mach–Zehnder Interferometer Model

The Mach–Zehnder interferometer is the most popular device [6, 20] to support Young’s double-slit experiment.

In Fig. 1a, a double path interferometer is shown. An input signal X under control function \( f \) causes Laser LS to emit the output signal \( \rho \) under BP (Bi-polarized filter) operation output a pair of signals: \( \rho^{ + } \) and \( \rho^{ - } \). Both signals are processed by SW output \( \rho_{L}^{ + } \) and \( \rho_{R}^{ - } \), and then IM to generate output signals IM(\( \rho_{L}^{ + } \), \( \rho_{R}^{ - } \)). In Fig. 1b, a representation model has been described with the same signals being used.

Fig. 1
figure 1

Double path model a Mach–Zehnder double path model, b Description model

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2.2 Emission and Absorption Measures of Quantum Interaction

Einstein established a model to describe atomic interaction [27,28,29,30] with radiation in 1916. For two-state systems, let a system have two energy states: the ground state \( E_{1} \) and the excited state \( E_{2} \). Let \( N_{1} \) and \( N_{2} \) be the average numbers of atoms in the ground and excited states, respectively. The numbers of states are changed from an emission state \( E_{2} \) to \( E_{1} \) with a rate \( \frac{{{\text{d}}N_{21} }}{{{\text{d}}t}} \), in the same time; the numbers of ground states are determined by absorbed energies from \( E_{1} \) to \( E_{2} \) with a rate \( \frac{{{\text{d}}N_{12} }}{{{\text{d}}t}} \), respectively. Let \( N_{12} \) be the number of atoms from \( E_{1} \) to \( E_{2} \) and \( N_{21} \) be the numbers from \( E_{2} \) to \( E_{1} \). In Einstein’s model, a measurement quaternion is \( \left\langle {N_{1} ,N_{2} ,N_{12} ,N_{21} } \right\rangle \).

CHSH proposed spin measures testing Bell inequalities [4, 6]. They applied \( \bot \to + \) and \( || \to - \) to establish a measurement quaternion

$$ \left\langle {N_{ + + } (a,b),N_{ + - } (a,b),N_{ - + } (a,b),N_{ - - } (a,b)} \right\rangle . $$

Experimental testing of Bell inequalities was performed by Aspect [5] in 1982. Four parameters are measured: transmission rate \( N_{t} \), reflection rate \( N_{r} \), correspondent rate \( N_{c} \), and the total number \( N_{\omega } \) in \( \omega \) time period. This set of measures is a quaternion \( \left\langle {N_{t} ,N_{r} ,N_{c} ,N_{\omega } } \right\rangle \). Among these, \( N_{c} \) is a new data type not in Einstein and CHSH methods, this parameter could be an extension of synchronous/asynchronous time-measurement.

3 Simulation Systems

3.1 Simulation Model

Using 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. This variant simulation system is composed of three components: Pre-process, Interaction, and Post-process, respectively, and shown in Fig. 2.

Fig. 2
figure 2

Variant simulation system; a Variant simulation system; b interactive component

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In Fig. 2a, three components of the variant simulation model are presented. At the pre-process, a \( N \) bit 0-1 vector X and a function \( f \) feed in to output a signal \( \rho \). After interactive component process, two groups of signals are the output: \( u \) for symmetry group and \( v \) for anti-symmetry group. In the post-process, all \( N \) bit vectors are processed by pre-processing and interactive components until all of the \( 2^{N} \) data set has been processed to transform symmetry and anti-symmetry signals into eight histograms: four for symmetry distributions and another four for anti-symmetry distributions.

In Fig. 2b, only the interaction component is selected, input signal \( \rho \) processed by BP to generate two signals \( \left\{ {\rho_{ - } ,\rho_{ + } } \right\} \). SW output triple signals \( \left\{ {\rho_{ - } ,1 - \rho_{ - } ,\rho_{ + } } \right\} \) though IM to generate two groups of signals \( u \) and \( v \).

3.2 Variant Principle

The variant principle is based on n-variable logic functions [31,32,33]. For any \( n \)-variables, \( x = x_{n - 1} \ldots x_{i} \ldots x_{0} , \) \( 0 \le i < n, \)\( x_{i} \in \{ 0,1\} = B_{2} \). Let a position \( j \) be the selected bit \( 0 \le j < n \), \( x_{j} \in B_{2} \) be the selected variable. Let output variable y and \( n \)-variable function \( f,\;y = f(x) \), \( y \in B_{2} ,x \in B_{2}^{n} \). For all states of \( x \), a set \( S(n) \) composed of the \( 2^{n} \) states can be divided into two sets: \( S_{ 0}^{j} (n) \) and \( S_{ 1}^{j} (n) \).

$$ \left\{ {\begin{array}{*{20}l} {S_{0}^{j} (n) = \left\{ {x|x_{j} = 0,\forall x \in B_{2}^{n} } \right\}} \hfill \\ {S_{1}^{j} (n) = \left\{ {x|x_{j} = 1,\forall x \in B_{2}^{n} } \right\}} \hfill \\ {S(n) = \left\{ {S_{0}^{j} (n),S_{1}^{j} (n)} \right\}} \hfill \\ \end{array} } \right. $$

For a given logic function \( f \), there are input and output pair relationships to define four meta-logic functions \( \left\{ {f_{ \bot } ,f_{ + } ,f_{ - } ,f_{T} } \right\} \):

$$ \left\{ {\begin{array}{*{20}c} {f_{ \bot } (x) = \left\{ {f(x)|x \in S_{0}^{j} (n),y = 0} \right\}} \\ {f_{ + } (x) = \left\{ {f(x)|x \in S_{0}^{j} (n),y = 1} \right\}} \\ {f_{ - } (x) = \left\{ {f(x)|x \in S_{1}^{j} (n),y = 0} \right\}} \\ {f_{T} (x) = \left\{ {f(x)|x \in S_{1}^{j} (n),y = 1} \right\}} \\ \end{array} } \right. $$

Two logic canonical expressions: AND-OR form is selected by \( \left\{ {f_{ + } (x),f_{T} (x)} \right\} \) as y = 1 items, and OR-AND form is selected from \( \left\{ {f_{ - } (x),f_{ \bot } (x)} \right\} \) as y = 0 items. Considering \( \left\{ {f_{T} (x),f_{ \bot } (x)} \right\} \), \( x_{j} = y \) items, they are invariant themselves.

To select \( \left\{ {f_{ + } (x),f_{ - } (x)} \right\} \); \( x_{j} \ne y \) forming variant logic expression. Let \( f(x) = \left\langle {f_{ + } |x|f_{ - } } \right\rangle \) be a variant logic expression. Any logic function can be expressed as a variant logic form. In \( \left\langle {f_{ + } |x|f_{ - } } \right\rangle \) structure, \( f_{ + } \) selected 1 item in \( S_{0}^{j} (n) \) as same as the AND-OR standard expression, and \( f_{ - } \) selecting relevant parts as same as the OR-AND expression 0 items in \( S_{1}^{j} (n) \). For a convenient understanding of variant representation, two-variable logic structures are illustrated for its 16 functions shown in Table 1.

Table 1 Two variable logic functions and variable logic representation (n = 2, j = 0)
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For example, checking two functions \( f \) = 3 and \( f \) = 12:

$$ \begin{array}{*{20}l} {\left\{ {f = 3: = \left\langle {0} \mathrel{\left | {\vphantom {0 3}} \right. \kern-0pt} {3} \right\rangle ,f_{ + } = 11:{ = }\left\langle {0} \mathrel{\left | {\vphantom {0 \phi }} \right. \kern-0pt} {\phi } \right\rangle ,f_{ - } = 2: = \left\langle {\phi } \mathrel{\left | {\vphantom {\phi 3}} \right. \kern-0pt} {3} \right\rangle } \right\}} \hfill \\ {\left\{ {f = 1 2: = \left\langle {2} \mathrel{\left | {\vphantom {2 1}} \right. \kern-0pt} { 1} \right\rangle ,f_{ + } = 14:{ = }\left\langle {2} \mathrel{\left | {\vphantom {2 \phi }} \right. \kern-0pt} {\phi } \right\rangle ,f_{ - } = 8: = \left\langle {\phi } \mathrel{\left | {\vphantom {\phi 1}} \right. \kern-0pt} { 1} \right\rangle } \right\}} \hfill \\ \end{array} $$

3.3 Variant Measures

Let \( \Delta \) be variant measure function [23, 33].

$$ \Delta = \left\langle {\Delta_{ \bot } ,\Delta_{ + } ,\Delta_{ - } ,\Delta_{T} } \right\rangle $$
$$ \begin{aligned} \Delta f(x) & = \left\langle {\Delta_{ \bot } f(x),\Delta_{ + } f(x),\Delta_{ - } f(x),\Delta_{T} f(x)} \right\rangle \\ & = \left\langle {\Delta f_{ \bot } (x),\Delta f_{ + } (x),\Delta f_{ - } (x),\Delta f_{T} (x)} \right\rangle \\ \end{aligned} $$
$$ \Delta f_{\alpha } (x) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {{\text{if}}\;f(x) = f_{\alpha } (x),\alpha \in \left\{ { \bot , + , - ,{\rm T}} \right\}} \hfill \\ {0,} \hfill & {\text{others}} \hfill \\ \end{array} } \right. $$

For any given n-variable state there is one position in \( \Delta f(x) \) to be 1 and other three positions are 0.

For any N bit 0-1 vector X; \( X = X_{N - 1} \ldots X_{J} \ldots X_{0} ,0 \le J < N,X_{\text{J}} \in \beta_{2} ,X \in \beta_{2}^{N} \) under n-variable function \( f \), n bit 0-1 output vector Y, \( Y = f(X) = \left\langle {f_{ + } |X|f_{ - } } \right\rangle , \) \( Y = Y_{N - 1} \ldots Y_{\text{J}} \ldots Y_{0} , \) \( 0 \le {\text{J}} < N, \) \( Y_{\text{j}} \in \beta_{2} , \) \( Y \in \beta_{2}^{N} \).

For the Jth position, be \( x^{\text{J}} = \left[ { \ldots X_{\text{J}} \ldots } \right] \in \beta_{2}^{n} \) to form \( Y_{\text{J}} = f(x^{\text{J}} ) = \left\langle {f_{ + } |x^{\text{J}} |f_{ - } } \right\rangle , \) let N bit positions be cyclic linked. Variant measures of \( f(X) \) can be decomposed

$$ \Delta \left\langle {X:Y} \right\rangle = \Delta f(X) = \sum\limits_{J = 0}^{N - 1} {\Delta f(x^{J} )} = \left\langle {N_{ \bot } ,N_{ + } ,N_{ - } ,N_{T} } \right\rangle $$

as a quaternion \( \left\langle {N_{ \bot } ,N_{ + } ,N_{ - } ,N_{T} } \right\rangle \).

For example, N = 10, given \( f \), \( Y = f({\text{X}}) \).

$$ \begin{array}{l@{\qquad}l@{\qquad}l@{\qquad}l@{\qquad}l@{\qquad}l@{\qquad}l@{\qquad}l@{\qquad}l@{\qquad}l@{\qquad}l@{\qquad}l} {\rm X} & = & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ {\rm Y} & = & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ \Delta({\rm X}:{\rm Y} ) & = & + & - & {\rm T} & \bot & + & - & {\rm T} & - & + & \bot \\ \end{array} $$
$$ \Delta f(X) = \left\langle {N_{ \bot } ,N_{ + } ,N_{ - } ,N_{T} } \right\rangle = \left\langle {2,3,3,2} \right\rangle ,N = 10 $$

Input and output pairs are 0-1 variables on the four combinations. For any given function \( f \), the quantitative relationship of \( \left\{ { \bot , + , - ,T} \right\} \) is determined directly from input/output sequences.

3.4 Measurement Equations

Using variant quaternion, signals are calculated by following equations. For any N bit 0-1 vector X, function \( f \), under \( \Delta \) measurement: \( \Delta f(x) = \left\langle {N_{ \bot } ,N_{ + } ,N_{ - } ,N_{T} } \right\rangle \), \( N = N_{ \bot } + N_{ + } + N_{ - } + N_{T} \) Signal \( \rho \) is defined by

$$ \rho = \frac{\Delta f(x)}{N} = \left\langle {\rho_{ \bot } ,\rho_{ + } ,\rho_{ - } ,\rho_{T} } \right\rangle $$
$$ \rho_{\alpha } = \frac{{N_{\alpha } }}{N},\;0 \le \rho_{\alpha } \le 1,\quad \alpha \in \left\{ { \bot , + , - ,T} \right\} $$

Using \( \left\{ {\rho_{ + } ,\rho_{ - } } \right\} \), a pair of signals \( \left\{ {u,v} \right\} \) are formulated:

$$ \left\{ {\begin{array}{*{20}c} {u = \left\langle {u_{0} ,u_{ + } ,u_{ - } ,u_{1} } \right\rangle = \left\{ {u_{\beta } } \right\}} \\ {v = \left\langle {v_{0} ,v_{ + } ,v_{ - } ,v_{1} } \right\rangle = \left\{ {v_{\beta } } \right\}} \\ \end{array} } \right. $$
$$ \beta \in \left\{ {0, + , - ,1} \right\} $$
$$ \left\{ {\begin{array}{*{20}l} {u_{0} = \rho_{ - } \oplus \rho_{ + } } \hfill \\ {v_{0} = (1 - \rho_{ - } )/2 \oplus (1 + \rho_{ + } )/2} \hfill \\ {u_{ + } = \rho_{ + } } \hfill \\ {v_{ + } = (1 + \rho_{ + } )/2} \hfill \\ {u_{ - } = \rho_{ - } } \hfill \\ {v_{ - } = (1 - \rho_{ - } )/2} \hfill \\ {u_{1} = \rho_{ - } + \rho_{ + } } \hfill \\ {v_{1} = (1 - \rho_{ - } + \rho_{ + } )/2} \hfill \\ \end{array} } \right. $$

where \( 0 \le u_{\beta } ,v_{\beta } \le 1 \), \( \beta \in \left\{ {0, + , - ,1} \right\} \), \( \oplus \): Asynchronous addition, +: Synchronous addition.

Using \( \left\{ {u,v} \right\} \) signals, each \( u_{\beta } \) (\( {\text{v}}_{\beta } \)) determines a fixed position in the relevant histogram to make vector X on a position. After complete \( 2^{N} \) data sequences, eight symmetry/anti-symmetry histograms of \( \left\{ {H(u_{\beta } |f)} \right\} \)\( \left( {\left\{ {H(v_{\beta } |f)} \right\}} \right) \)\( \beta \in \left\{ {0, + , - ,1} \right\} \) are generated.

4 Simulation Results

The simulation provides a series of output results. In this section, two cases are selected: \( N = \left\{ {12,13} \right\} \), \( n = 2,j = 0 \), \( \left\{ {f = 3,f_{ + } = 11,f_{ - } = 2} \right\} \), and \( \left\{ {f = 12,f_{ + } = 14,f_{ - } = 8} \right\} \). Corresponding to double path, right path, left path, symmetric and nonsymmetric conditions, respectively. For the convenience of comparison, sample cases are shown in Fig. 3a–c. In Fig. 3a, representation patterns are illustrated. Figure 3b represents \( f \) = 3 conditions and Fig. 3c represents \( f \) = 12 conditions, respectively. Eight histograms of \( H(u_{ + } |f) = H(u_{ - } |f) \) are shown with results represented by symmetric meta-functions in four groups.

Fig. 3
figure 3figure 3

Results of symmetric meta distributions

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5 Analysis of Results

5.1 Visual Distributions

In \( H(u_{ + } |f) = H(u_{ - } |f) \) conditions, \( \left\{ {} \right.H(u_{1} |f),H(v_{1} |f)\left. {} \right\} \) have significant interference patterns different from other conditions. Output results are balanced.

5.2 Particle Statistical Distributions

For all symmetric or nonsymmetric cases under \( \oplus \) asynchronous addition operations, relevant values meet \( 0 \le u_{0} ,v_{0} ,u_{ - } ,v_{ - } ,u_{ + } ,v_{ + } \le 1 \).

Checking \( \left\{ {} \right.H(u_{0} |f),H(v_{0} |f)\left. {} \right\} \) series, \( \left\{ {H(u_{ + } |f),H(u_{ - } |f)} \right\} \) and \( \left\{ {} \right.H(v_{ + } |f),H(v_{ - } |f)\left. {} \right\} \) satisfy the following equation:

$$ \left\{ {\begin{array}{*{20}c} {H(u_{0} |f) = H(u_{ - } |f) + H(u_{ + } |f)} \\ {H(v_{0} |f) = H(v_{ - } |f) + H(v_{ + } |f)} \\ \end{array} } \right. $$

The equation is true even N and n in different values.

5.3 Wave Interference Patterns

Different interference properties are observed clearly in \( H(u_{ + } |f) = H(u_{ - } |f) \) and \( H(v_{ + } |f) = H(1 - v_{ - } |f) \) conditions. Under + synchronous addition operations, relevant values meet \( 0 \le u_{1} ,v_{1} ,u_{ - } ,v_{ - } ,u_{ + } ,v_{ + } \le 1 \).

Checking \( \left\{ {} \right.H(u_{1} |f),H(v_{1} |f)\left. {} \right\} \) distributions especially in Fig. 3b–c \( \left\{ {} \right.u{}_{1},v_{1} \left. {} \right\} \) cases extremely strong interferences appeared and compared with \( \left\{ {} \right.H(u_{ + } |f),H(u_{ - } |f)\left. {} \right\} \) and \( \left\{ {} \right.H(v_{ + } |f),H(v_{ - } |f)\left. {} \right\} \), there are significant differences. Spectra in different cases illustrate wave interference properties. From listed histogram distributions, they are all satisfied

$$ \left\{ {\begin{array}{*{20}c} {H(u_{1} |f) \ne H(u_{ - } |f) + H(u_{ + } |f) = H(u_{0} |f)} \\ {H(v_{1} |f) \ne H(v_{ - } |f) + H(v_{ + } |f) = H(v_{0} |f)} \\ \end{array} } \right. $$

Single and double peaks are shown in interference patterns as classical double-slit distributions.

5.4 Quaternion Measures

It is interesting to see the relationship between the variant quaternion and other measures.

In the variant quaternion, \( \Delta f(x) = \left\langle {N_{ \bot } ,N_{ + } ,N_{ - } ,N_{T} } \right\rangle \), \( N = N_{ \bot } + N_{ + } + N_{ - } + N_{T} \).

In Einstein’s two-state system of interaction \( \left\langle {N_{1} ,N_{2} ,N_{12} ,N_{21} } \right\rangle \) allows the following equations to be established:

$$ \left\{ {\begin{array}{*{20}l} {N_{1} = N_{ \bot } + N_{ + } } \hfill \\ {N_{2} = N_{ - } + N_{T} } \hfill \\ {N_{12} = N_{ + } } \hfill \\ {N_{21} = N_{ - } } \hfill \\ {N = N_{1} + N_{2} } \hfill \\ \end{array} } \right. $$

From the equations, the measured pair \( \left\{ {} \right.N_{21} ,N_{12} \left. {} \right\} \) has a 1-1 correspondence to \( \left\{ {} \right.N_{ - } ,N_{ + } \left. {} \right\} \).

Selecting \( + \to 1 \), \( - \to 0 \), CHSHs \( N_{ \pm , \mp } (a,b) \) measures meet

$$ \left\{ {\begin{array}{*{20}c} {N_{ + , + } (a,b) \to N_{T} } \\ {N_{ + , - } (a,b) \to N_{ - } } \\ {N_{ - , + } (a,b) \to N_{ + } } \\ {N_{ - , - } (a,b) \to N_{ \bot } } \\ \end{array} } \right. $$
$$ (N_{ + + } ,N_{ + - } ,N_{ - + } ,N_{ - - } ) \to (N_{T} ,N_{ - } ,N_{ + } ,N_{ \bot } ), $$

Let \( N = N_{ + + } + N_{ + - } + N_{ - + } + N_{ - - } \), CHSH quaternion is a permutation of the variant quaternion.

Aspect’s quaternion \( (N_{t} ,N_{r} ,N_{c} ,N_{\omega } ) \) have the following corresponding:

$$ \left\{ {\begin{array}{*{20}c} {N_{t} \to N_{ - } } \\ {N_{r} \to N_{ + } } \\ {N_{\omega } \to N_{{}} } \\ \end{array} } \right. $$

There is no parameter in the variant quaternion for the parameter \( N_{c} \). It indicates joined action numbers to distinguish single and double paths, corresponding to \( \left\{ {} \right.u{}_{0},v_{0} \left. {} \right\} \) and \( \left\{ {} \right.u{}_{1},v_{1} \left. {} \right\} \) times. In an actual experiment, this parameter is significant. In a simulated system, the parameter is a control coefficient that separates two types of measured paths \( \left\{ {} \right.u{}_{0},v_{0} \left. {} \right\} \) and \( \left\{ {} \right.u{}_{1},v_{1} \left. {} \right\} \) in the integration of comparisons on real experiments.

6 Conclusions

Analyzing N bit 0-1 vector and its exhaustive sequences for variant measurement, this system simulates double path interference properties through different accurate distributions. Using this model, two groups of parameters \( \left\{ {} \right.u_{\beta } \left. {} \right\} \) and \( \left\{ {} \right.v_{\beta } \left. {} \right\} \) describe the left path, right path, double paths for particle, and double path for wave with distinguished symmetry and anti-symmetry properties.

Only synchronous conditions, double path system provides wave-like interference patterns different from classical ones.

Compared with the variant quaternion and other quaternion structures, it is helpful to understand possible properties of usages and limitations for variant simulation systems.

The complexity of n-variable function space has a size of \( 2^{{2^{n} }} \). Whole simulation complexity is determined by \( O(2^{{2^{n} }} \times 2^{N} ) \) 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.