Theory of Quantum Path Entanglement and Interference with Multiplane Diffraction of Classical Light Sources

MPD setup is formed from $N-1$ diffraction planes of multiple slits in front of a classical light source and the measurement of interference pattern with N sensor planes, i.e., both diffraction and sensing on the same plane, as shown in Figure 1a. It is also possible to locally count the diffracted photons with the measurement planes inserted between the diffraction planes as discussed in Section 2.6. The utilized light source is assumed to be coherent as the closest analog of a classical light field emphasizing the absence of nonclassical states of light such as single photon generation, squeezed light, or multiple particles of entangled photons [25]. The standard laser output is almost perfectly a coherent state corresponding to the fundamental transverse modes of light field distribution producing Gaussian beams. This coherent Gaussian wave function keeps the position and momentum uncertainties stationary as emphasized by Glauber [26]. It is an eigenstate of the annihilation operator $\widehat{a}$ for the harmonic oscillator, i.e., $\widehat{a}|\alpha \rangle =\alpha |\alpha \rangle$, represented as follows in the complete orthonormal basis of the number states $|n\rangle$ of the single mode oscillator [26]: where its representation in the position basis gives the Gaussian form. Therefore, the source is assumed to have normalized Gaussian wave function ${\Psi }_0(x_0)\equiv \exp \left(-x_0^2/(2{\sigma }_0^2)\right)/\sqrt{{\sigma }_0/\sqrt{\pi }}$ with the standard deviation term ${\sigma }_0$.

Each plane is assumed to be capable of performing measurement with photodetectors for counting the number of photons hitting the detector area. Therefore, a plane either allows projective diffraction of light through slits denoted by the operator symbol $\mathbf{P}$ or performs measurement denoted by $\mathbf{M}$ on its sensor array positions where there are no slits. Gaussian slits are utilized with FPI modeling for simplicity [2,8] as mathematically described in Equation (7) in the next subsection. Light is assumed to perform free space propagation between consecutive planes. The plane with the index j has $S_j$ slits, where the central positions and widths of slits are denoted by $X_{j,i}$ and $W_{j,i}$, respectively, and $j\in [1,N-1]$ and $i\in [1,S_j]$. The widths of the slits are assumed to be the same on each plane but not constrained among different planes. Distance between the ith and jth planes is denoted by $L_{i,j}$, where the distance from the light transmitter source to the first plane is given by $L_{0,1}$. Light is assumed to have propagation in the z-axis with the velocity given by c, while quantum superposition interference is observed in the x-axis as a one-dimensional model which can be easily extended to two dimensions (2D) [8]. Interplane distances and durations are denoted by the vectors ${\overrightarrow{L}}^T=[L_{0,1}\dots L_{N-1,N}]$ and ${\overrightarrow{t}}^T=[t_{0,1}\dots t_{N-1,N}]\equiv {\overrightarrow{L}}^T/c$, respectively, where transpose is denoted by ${(.)}^T$. The value ${\overrightarrow{t}}^T$ is accurate with the assumption $L_{j-1,j}\gg W_{j,i},X_{j,i}$ for $j\in [1,N-1]$ and $i\in [1,S_j]$ such that QM effects are emphasized in the x-axis. Nonrelativistic modeling is assumed. We do not consider the effects of environment dephasing or decohering of the interference pattern for double-slit setups [27,28]. Furthermore, minor effects of exotic paths [29] on the numerical results are ignored as discussed in Reference [8] without affecting the main modeling.

Free-particle evolution kernel for the optical propagation paths between time–position values $(t_j,x_j)$ and $(t_{j+1},x_{j+1})$ is defined as follows [9,10] with the same form for electron propagation [2,8]: where $\Delta t=t_{j+1}-t_j$, $\Delta x=x_{j+1}-x_j$, $m\equiv \hslash k/c$ is the virtual mass term for the photon with the wave number $k=2\pi /\lambda$, and $\lambda$ is the wavelength of the light.

The validity of Fresnel diffraction formulation for quantum optical propagation is verified based on recent experimental [30] and theoretical [31] studies, while Fourier optics [32] extension of MPD is recently proposed in Reference [9]. Therefore, the Fresnel diffraction integral for free space proposed in Equation (4) and its consecutive application with FPI formalism are theoretically valid and highly reliable for the simple design of MPD. The proposed theoretical model significantly promises to be verified with near-future experiments due to the simplicity of the setup. Then, the propagated wave function $|{\Psi }_j\rangle ={\int }_{-\infty }^{\infty }\mathrm{d}{x}_j|x_j\rangle {\Psi }_j(x_j)$ on the jth plane becomes as follows by utilizing Equation (4) consecutively in FPIs [8,9,10]: where ${\psi }_{j,n}(x_j)$ is the contribution for each nth propagation path through the slits on the overall superposition, and the definitions of the notations n and $N_j$ are explained next while ${\mathrm{Y}}_j={\chi }_0\left({\prod }_{l=1}^{j-1}\sqrt{{\xi }_l}\right)$; the constants $A_{j-1}$, $B_{j-1}$, ${\chi }_0$, and ${\xi }_l$ for $l\in [1,j-1]$; ${\mathbf{H}}_{j-1}={\mathbf{H}}_{R,j-1}+ı{\mathbf{H}}_{I,j-1}$; and the vectors ${\overrightarrow{c}}_{j-1}$ and ${\overrightarrow{d}}_{j-1}$ depending on the group of $\{\hslash$, m, ${\sigma }_0$, $t_{l,l+1}$, and ${\beta }_l\}$ for $l\le j-1$ are explicitly defined in Reference [8]. Explicit forms of the parameters required for double- and triple-plane setups are provided in Section 4 while formulating LGIs and QPI, respectively, in the following discussions. The total number of paths just before diffraction on the jth plane is calculated by $N_j={\prod }_{l=1}^{j-1}{S}_l$, while the set of slit positions for the path indexed with $n\in [0,N_j-1]$ is denoted by ${\overrightarrow{x}}_n\equiv {[X_{1,s_{n,1}}{X}_{2,s_{n,2}}\dots X_{j-1,s_{n,j-1}}]}^T$ while each nth path is indexed by the set of diffracted slits as the following: where the specific slit on lth plane for nth trajectory is indexed with $s_{n,l}$. The same symbol of the position vector $\overrightarrow{x}$ is used for both the dimensions N and j. The size of the vector is inferred from the index of the current plane analyzed throughout the text. The position on the jth plane is denoted by $x_j$. In Equation (5) for $j=N$, each path reaching the Nth plane is indexed by n for $n\in [0,N_p-1]$ as shown in Figure 1b for a simple example of $N_p=4$, where total number of paths is given by multiplying the number of slits on each plane as $N_p\equiv {\prod }_{j=1}^{N-1}{S}_j$. The vector ${\overrightarrow{x}}_n\equiv$$[X_{1,s_{n,1}}{X}_{2,s_{n,2}}$$\dots$$X_{N-1,s_{n,N-1}}{]}^T$ denotes the set of slit positions ordered with respect to the plane indices for nth path for the case of N planes. Next, diffraction and measurement operators are theoretically defined by emphasizing the operator algebra of multiplane evolution.

Projection operator denotes the light to be in the Gaussian slit in a coarse-grained sense [8,33] as follows: where $g_{j,i}(x_j)\equiv \exp \left(-{(x_j-X_{j,i})}^2/(2{\beta }_{j,i}^2)\right)$ is the slit projection function and the effective slit width is $W_{j,i}\equiv 2\sqrt{2}{\beta }_{j,i}$, i.e., leading to a $1/e^2$ drop in the intensity, where $j\in [1,N-1]$ and $i\in [1,S_j]$. Projectors are mutually exclusive with high accuracy such that slit distances are chosen large enough to satisfy $\exp \left(-{(X_{j,i_1}-X_{j,i_2})}^2/(2{\beta }_{j,i_1}^2)\right)\ll 1$ for $i_1\ne i_2$. Total diffraction through all slits of the jth plane has the operator ${\mathbf{P}}_j\equiv {\sum }_{i=1}^{S_j}{\mathbf{P}}_{j,i}$. Measurement operators are redefined due to the proposed Gaussian slit design such that trace preserving equality is satisfied, i.e., ${\mathbf{M}}_j^{†}{\mathbf{M}}_j+{\mathbf{P}}_j^{†}{\mathbf{P}}_j=\mathbf{I}$, where $\mathbf{I}$ is the identity operator and ${(.)}^{†}$ or ${(.)}^H$ denotes Hermitian or conjugate transpose operation. It is assumed that wave function at time $t=t_0$ evolves to $|{\Psi }_j\rangle$ and $|{\Psi }_j^{+}\rangle$ for just before and just after diffraction on the jth plane at $t_j^{-}$ and $t_j^{+}$, respectively. The state of the light at $t_j^{+}$ has experienced either ${\mathbf{M}}_j$ or ${\mathbf{P}}_j$. The measurement operator on the jth plane is defined as the following: Therefore, if we define the measurement operator in FPI formalism as multiplication of the wave function with $m_j(x_j)$ reducing the probability to measure the light while approaching the slit center, then the following is obtained by using Equation (8):

There are two different types of detection mechanisms in MPD design denoted by $Re{c}_1$ and $Re{c}_2$. In $Re{c}_1$, all of the planes for $j\in [1,N]$ have detectors measuring the incident light and $Re{c}_1$ is the model proposed in this article forming a complete set of diffractive projection ${\mathbf{P}}_j$ and measurement ${\mathbf{M}}_{\tilde{j}}$ operators until the final detector plane N for $j\in [1,N-1]$ and $\tilde{j}\in [1,N]$. In this article, $Re{c}_1$ modeling is utilized to model history-based time evolution of the light. An example is shown in Figure 1b, where there is a total of seven different sets of consecutive events forming a complete set of histories. The proposed setup is modeled compatible with the consistent histories approach defined in Reference [3] or the entangled histories framework in Reference [1]. On the other hand, the receiver type with the sensors only on the final plane is denoted by $Re{c}_2$. In $Re{c}_2$, i.e., the modeling utilized in Reference [8] for QC, only the final intensity distribution or interference pattern on the detector plane is measured. There is either no detection at the time $t_N^{+}$ or the light is detected on the final detector plane with the index N. An operator denoting no detection is defined as ${\mathbf{M}}_o$ to form a complete set for $Re{c}_2$; then, ${\mathbf{M}}_N^{†}{\mathbf{M}}_N+{\mathbf{M}}_o^{†}{\mathbf{M}}_o=\mathbf{I}$. Next, consistent histories approach is applied for MPD setup.

Following the definition of consistent histories [3,4,5] and entangled histories [1], a history state is defined for MPD based on the set of projections ${\mathbf{M}}_j$ and ${\mathbf{P}}_{j,i}$ on each jth plane for $j\in [1,N]$ and $i\in [1,S_j]$. History Hilbert space is defined as follows: where ${\mathcal{H}}_j$ denotes the set of projections on planes and ⊙ denotes tensor product operation. Hilbert space until $t_j^{+}$ includes both projections ${\mathbf{P}}_j$ and ${\mathbf{M}}_l$ on the planes with the indices $l\le j$ since the light is detected at some plane until $t_j$ or still diffracting through the jth plane. A general history state with QPE composed of superposition of trajectories is denoted as follows based on the notation (similar to bra-ket but with different notations of $(.|$ and $|.)$ for the histories corresponding to $\langle .|$ and $|.\rangle$, respectively) in Reference [1]: where $|{\Psi }_j)$ is some history state between times $t_0$ and $t_j$ for $t_j>t_0$, the projector $\left[{\mathbf{O}}_n(t_j)\right]$ denotes either of ${\mathbf{M}}_l$ or ${\mathbf{P}}_{l,i}$ for $l\le j$ and $i\in [1,S_l]$, and ${\pi }_n$ as 0 or 1 is some permutation choosing a compound set of histories indexed by n. Observe that $t_j$ includes measurements ${\mathbf{M}}_l$ for $l\le j$ as possible events such that the state does not change after measurement. It also includes events with zero probability such as jth plane projection at times not equal to $t_j$. Some examples for $N=4$ are as follows: The state $|{\Psi }_4^a)$ shows that the light is detected on the first plane at $t_1$ while not changing at consecutive time states, i.e., without diffracting even from the first plane. In $|{\Psi }_4^b)$, the light is diffracted from the first slit of the first plane at $t_1$, then is diffracted from the fourth slit of the second plane at $t_2$ and from the second slit of the third plane at $t_3$, and is finally measured on the fourth plane. The third example $|{\Psi }_4^c)$ is a state with zero probability due to the orthogonality of the operators on different planes. A simple example for three planes with two slits is shown in Figure 1b with seven different history states while $N_p=4$ of them reach the final detector plane as consecutively diffracted trajectories. History Hilbert space summing to the identity denoted by ${\overline{I}}_H$ as the family based upon an initial state and neglecting the histories with zero probability is described as follows [3]: where ${\left[{\mathbf{M}}_j\right]}^{\odot \alpha }$ denotes $\alpha \equiv N+1-j$ consecutive measurements of $\left[{\mathbf{M}}_j\right]$ on the same plane. This includes all the possible history states and evolution for the light until $t=t_N$ starting from $t_0$. A chain operator is presented in Reference [1] to define the inner product between history states which maps a history state to an operator. The chain operator provides history states with positive semi-definite inner products. This operator is inherently defined in the MPD system as the free-particle evolution kernel $K(x_1,t_1;x_0,t_0)$. Assume that the free-particle evolution operator with the notation $U_{j+1,j}$ acts as the bridging operator connecting projections at times $t_j$ and $t_{j+1}$. Then, chain operator denoted by ${\chi }_{t_{j+1},t_j}$ for the time duration ($t_j$, $t_{j+1}$) is defined as follows: where $\left[{\mathbf{O}}_n(t_{j+1})\right]$ and $\left[{\mathbf{O}}_n(t_j)\right]$ in $\stackrel{4}{=}$ denote the cases which are not presented in the first three definitions. $\mathit{I}$ is the identity operator equalizing the consecutive measurements on the same plane, i.e., ${\mathbf{M}}_j^l={\mathbf{M}}_j$, for any integer l. Furthermore, it bridges dynamically not possible history states which have zero probability to occur as discussed in Reference [3]. These include consecutive measurements on different planes such as $\left[{\mathbf{M}}_{j+1}\right]\odot \left[{\mathbf{M}}_j\right]$, future projection or measurements at a previous time such as $\left[{\mathbf{M}}_j\right]\odot \left[{\mathbf{P}}_{j+1}\right]$, or consecutive sets of the same projector ${\mathbf{P}}_j$ at future times such as $\left[{\mathbf{P}}_j\right]\odot \left[{\mathbf{P}}_j\right]$, where free-space propagation in the z-axis prevents this. Then, the compound history state mapped or affected by the chain operator is defined as follows: where $V_{j+1,j}$ denotes either $U_{j+1,j}$ or $\mathit{I}$.

Besides that, MPD allows to model and explore varying kinds of superposition of history states and QPEs similar to the specific entangled states discussed in Reference [1] resembling the temporal counterpart of Bell states. For example, entangled history states of the GHZ type is experimentally tested in Reference [11]. It is an open issue to utilize MPD to generate and test such states with important implications and applications based on QPE. Next, probability amplitudes of histories are modeled.

LGIs test the temporal correlations by measuring at different times in analogy with spatial Bell’s inequalities for the entanglement between spatially separated systems [12,13]. Three-time correlation-based inequality is defined in Equation (2) as $K=C_{01}+C_{12}-C_{02}\le 1$, where the bound is violated quantum mechanically with dichotomic systems, i.e., $Q_j=\pm 1$ for $j\in [0,2]$, reaching the bound $3/2$ for a two-level system with the maximum LGI violation of $1/2$. $C_{j_1,j_2}\equiv 〈Q_{j_1}{Q}_{j_2}〉$ is the expected value of the multiplication of the dichotomic observables, which is equal to $C_{j_1,j_2}={\sum }_{j_1,j_2}p(j_1,j_2)Q_{j_1}{Q}_{j_2}$, where $p(j_1,j_2)$ is the probability for the measurement of $Q_{j_1}$ and $Q_{j_2}$ at times $t_{j_1}$ and $t_{j_2}$, respectively, and $t_2>t_1>t_0$. Noninvasiveness or non-disturbing structure of the measurements should be clearly satisfied in order to reduce the “clumsiness loophole” [15,16], i.e., experimental limitations and disturbance of the clumsy measurements making it difficult to convince a macrorealist. In Reference [16], ambiguous measurements are utilized to revise Equation (2) by including the effect of signaling. In this article, the same formulation is extended for the MPD setup exploiting simple architecture of slits.

The correlation and entanglement in time are tested with the two-plane setup, where each plane includes triple slits as shown in Figure 2a. It is assumed that the light diffracting through the first plane is taken into account while calculating probability amplitudes, i.e., utilizing negative measurement techniques. For example, if the measured state is set to ${\mathbf{P}}_{1,1}$, then the second and third slits are closed, forcing the light to diffract through only the first slit setting the measurement result. Furthermore, denote $p_1(i_{1,1})\equiv Pro{b}_{1,i_{1,1}}^P$ and $p_1(\{i_{1,1},i_{1,2}\})\equiv Pro{b}_{1,i_s}^P$, where $i_s=\{i_{1,1},i_{1,2}\}$ for $i_{1,1},i_{1,2}\in [1,3]$ and $i_{1,1}\ne i_{1,2}$. The probability $p_1(\{i_{1,1},i_{1,2}\})$ corresponds to the measurement result for ${\mathbf{P}}_{1,i_{1,1}}\cup {\mathbf{P}}_{1,i_{1,2}}$ being projected in one of the slits with the indices $i_{1,1}$ and $i_{1,2}$ on the first plane. Similarly, $p_1(\{1,2,3\})$ denotes the overall projection on superposition in all three slits. On the other hand, assume that $p_{1,2}(\{i_{1,1},i_{1,2}\},{\widehat{i}}_2)$ denotes the probability for the history: where $\left[{\mathbf{O}}_{{\widehat{i}}_2}(t_2)\right]$ is one of $\left[{\mathbf{P}}_{2,1}\right]$, $\left[{\mathbf{P}}_{2,2}\right]$, $\left[{\mathbf{P}}_{2,3}\right]$, or $\left[{\mathbf{M}}_2\right]$ denoted by ${\widehat{i}}_2=1,2,3$, and 4, respectively. Similar to Equations (26) and (27), $p_{1,2}(\{i_{1,1},i_{1,2}\},{\widehat{i}}_2)$ for ${\widehat{i}}_2\in [1,3]$ is found as follows: where the elementary wave function is found with Equation (5) by using i as the path index as follows: where ${\Gamma }_1=A_1+ıB_1$, $H_1=H_{R,1}+ıH_{I,1}$, $r_1=c_1+ıd_1$, $i\in [1,3]$, ${\chi }_0$, ${\xi }_1$, $A_1$, $B_1$, $H_{R,1}$, $H_{I,1}$, $c_1$, and $d_1$ are defined in Section 4. Similarly, $p_{1,2}(\{i_{1,1},i_{1,2}\},4)$ is defined as follows: where $i_{1,1},i_{1,2}\in [1,3]$ and $i_{1,1}\ne i_{1,2}$. $p_{1,2}(i_{1,1},{\widehat{i}}_2)$, and $p_{1,2}(\{1,2,3\},{\widehat{i}}_2)$ denote the probabilities for $\left[{\mathbf{O}}_{{\widehat{i}}_2}(t_2)\right]\odot \left[{\mathbf{P}}_{1,i_{1,1}}\right]\odot \left[{\rho }_0\right]$ and $\left[{\mathbf{O}}_{{\widehat{i}}_2}(t_2)\right]\odot \left(\left[{\mathbf{P}}_{1,1}\right]+\left[{\mathbf{P}}_{1,2}\right]+\left[{\mathbf{P}}_{1,3}\right]\right)\odot \left[{\rho }_0\right]$, where $i_{1,1}\in [1,3]$ and ${\widehat{i}}_2\in [1,4]$. The same formulation is valid also for the second plane for $p_2(i_{2,1})$, $p_2(\{i_{2,1},i_{2,2}\})$, and $p_2(\{1,2,3\})$ for $i_{2,1},i_{2,2}\in [1,3]$ and $i_{2,1}\ne i_{2,2}$. Observe that, at time $t_2$, it is assumed that ${\mathbf{M}}_2$ is also included in calculations providing a complete set $\mathbf{I}={\mathbf{M}}_2+{\sum }_{i=1}^3{\mathbf{P}}_{2,i}$. Negative measurement methodology for the first plane is utilized such that the light only diffracting through the first plane is utilized in calculating probabilities. Therefore, all the probability calculations based on Equations (26), (27), (29), and (31) are normalized by ${\Gamma }_c\equiv ({\sum }_{i=1}^{3}Pro{b}_{1,i}^P{)}^{-1}$. The probabilities denoted by $p_j(i_{j,1})$, $p_j(\{i_{j,1},i_{j,2}\})$, $p_j(\{1,2,3\})$ for $j\in [1,2]$, $p_{1,2}(i_{1,1},{\widehat{i}}_2)$, $p_{1,2}(\{i_{1,1},i_{1,2}\},{\widehat{i}}_2)$, and $p_{1,2}(\{1,2,3\},{\widehat{i}}_2)$ are assumed to be normalized through the rest of the article. The normalized operator is defined as ${\mathbf{P}}_{1,i}^N\equiv {\Gamma }_c{\mathbf{P}}_{1,i}$ for $i\in [1,3]$.

Assume that an ambiguous measurement set of three projections composed by $\left[{\mathbf{O}}_1^A(t_1)\right]\equiv \left[{\mathbf{P}}_{1,1}^N\right]+\left[{\mathbf{P}}_{1,2}^N\right]$, $\left[{\mathbf{O}}_2^A(t_1)\right]\equiv \left[{\mathbf{P}}_{1,1}^N\right]+\left[{\mathbf{P}}_{1,3}^N\right]$, and $\left[{\mathbf{O}}_3^A(t_1)\right]\equiv \left[{\mathbf{P}}_{1,2}^N\right]+\left[{\mathbf{P}}_{1,3}^N\right]$ is defined. The setups for ambiguous measurements are shown in Figure 2b–d, respectively. In addition, an assignment of dichotomic indices for the measurement results is designed denoted by $Q_{1,i_1^A}\equiv \pm 1$ and $Q_{2,{\widehat{i}}_2}\equiv \pm 1$ for $\left[{\mathbf{O}}_{i_1^A}^A(t_1)\right]$ and $\left[{\mathbf{O}}_{{\widehat{i}}_2}(t_2)\right]$, respectively, where $i_1^A\in [1,3]$ and ${\widehat{i}}_2\in [1,4]$. $Q_0\equiv 1$ denotes initial condition $\left[{\rho }_0\right]$ with unity probability. These dichotomic indices can be assigned arbitrarily while they are chosen in Section 2.6 based on the maximization of LGI violation by comparing all the possible assignment combinations. Then, utilizing a similar architecture to the ambiguous LGI, i.e., Equation (14) in Reference [16], a conversion matrix $\mathbf{D}$ is defined inferring the probability $p_1(i_{1,1})$ from the ambiguous measurements with ${\widehat{p}}_1(i_{1,1})\equiv {\sum }_{i_1^A}{D}_{i_{1,1},i_1^A}{p}_1^A(i_1^A)$, where $p_1^A(i_1^A)$ denotes the probability for the history $\left[{\mathbf{O}}_{i_1^A}^A(t_1)\right]\odot \left[{\rho }_0\right]$ for $i_1^A\in [1,3]$, $D_{i_{1,1},i_1^A}$ is the element at the $i_{1,1}$th row and the $i_1^A$th column of the conversion matrix $\mathbf{D}$, and ${\widehat{p}}_1(i_{1,1})$ denotes the inferred probability such that a macrorealist will not observe any problem. Similarly, ${\widehat{p}}_{1,2}(i_{1,1},{\widehat{i}}_2)$ becomes the following: where $p_{1,2}^A(i_1^A,{\widehat{i}}_2)$ denotes the probability for the history $\left[{\mathbf{O}}_{{\widehat{i}}_2}(t_2)\right]\odot \left[{\mathbf{O}}_{i_1^A}^A(t_1)\right]\odot \left[{\rho }_0\right]$, where $i_{1,1},i_1^A\in [1,3]$ and ${\widehat{i}}_2\in [1,4]$, while it is found in Equations (29) and (31) by normalizing as follows: where $i_{1,1}^A$ and $i_{1,2}^A$ correspond to the the event $\left[{\mathbf{O}}_{i_1^A}^A(t_1)\right]$, i.e., $\{i_{1,1}^A,i_{1,2}^A\}$ equals $\{1,2\}$, $\{1,3\}$, and $\{2,3\}$ for $i_1^A\equiv 1$, 2, and 3, respectively. For example, for the proposed setup, ${\widehat{p}}_1(1)=(p_1^A(1)+p_1^A(2)-p_1^A(3))/2$ since the following probability relation holds: Therefore, $D_{11}=0.5$, $D_{12}=0.5$, and $D_{13}=-0.5$. Similarly, $D_{21}=D_{23}=D_{32}=D_{33}=0.5$ and $D_{22}=D_{31}=-0.5$. Then, a macrorealist is convinced that the inferred probabilities are utilized for the calculations of $C_{01}$, $C_{12}$, and $C_{02}$ with Equation (2) by replacing $p_{1,2}(i_{1,1},{\widehat{i}}_2)$ with ${\widehat{p}}_{1,2}(i_{1,1},{\widehat{i}}_2)$ and by properly defining the degree of signaling level between the first and second planes for the ambiguous measurements increasing the required LGI bound. Then, following the similar methodology in Reference [16] (Equations (5) and (14) in Reference [16]), $K=C_{01}+C_{12}-C_{02}$ with $Q_0=1$ is easily transformed into the combination of the standard LGI term free of the invasive measurement and a signaling term as follows by firstly replacing the measured probabilities with the inferred ones and then by inserting into K: where the first term is the standard LGI definition with inferred probabilities and the second term includes inferred signaling terms ${\Delta }_S({\widehat{i}}_2)$ between the first and second planes for the measurement $\left[{\mathbf{O}}_{{\widehat{i}}_2}(t_2)\right]$ for each ${\widehat{i}}_2$. It is modeled as a signaling quantifier showing the influence of the measurement at time $t_1$ to the measurement at time $t_2$ and is defined by utilizing ambiguous measurements as follows: where $\left({\sum }_{i_{1,1}=1}^3{D}_{i_{1,1},i_1^A}\right)=1/2$. Therefore, the no-signaling-in-time (NSIT) condition for the definition with ambiguous measurement of ${\Delta }_S({\widehat{i}}_2)=0$ is expected to convince a macrorealist about the reliability of the measurement setup. Then, the violation of LGI free of the invasive measurement becomes the following: where the summation at the right side of the inequality, i.e., $K_V-1$, shows the invasiveness of the measurements and the signaling level. Therefore, measured values $p_{1,2}^A(i_1^A,{\widehat{i}}_2)$ are utilized to check the violation compatible with respect to the objections of a macrorealist. If Equations (29), (31), and (33) are inserted into Equations (35), (36), and (38), then the following is obtained: where ${\overrightarrow{l}}_1={[11-1]}^T$; ${\overrightarrow{l}}_2={[1-11]}^T$; ${\overrightarrow{l}}_3={[-111]}^T$; the functions $f_1(.)$, $f_2(.)$, $f_{1,2}(.)$, and $f_V(.)$; and the variables $f_T$, $G_1$, and $G_2$ are defined in Section 4. Next, quantum interference among the paths, i.e., QPI, is defined for the MPD setup.