Forward-backward stochastic simulations: Q-based model for measurement and Bell-nonlocality consistent with weak local realistic premises

TL;DR

This work introduces a Q-function based forward-backward stochastic framework to model quantum measurement and entanglement, showing how macroscopic realism and no-signaling can coexist with Bell nonlocality. By representing the quantum state with a positive phase-space distribution $Q(x,p,t)$ and solving forward-backward stochastic trajectories, the authors derive Born’s rule from amplification-induced branch formation and reveal a postselected “hidden loop” structure that couples forward and backward paths. The approach yields a causal account of measurement, projection, and collapse that preserves no-signaling while explaining EPR correlations and CV Bell nonlocality through deterministic and probabilistic hidden-variable descriptions, including weak forms of local realism. The results demonstrate that Bell violations emerge from interference terms that become observable only after joint setting changes, avoiding genuine retrocausality and offering a unified interpretation of measurement, decoherence-like amplification, and nonlocal correlations with potential experimental tests. Overall, the paper provides a comprehensive, causally consistent framework linking phase-space Q-functions, FBSE dynamics, and weak realism premises to address foundational questions in quantum measurement and entanglement.

Abstract

We show how measurement and nonlocality can be explained consistently with macroscopic realism and no-signaling, and causal relations for macroscopic quantities. Considering measurement of a field amplitude $\hat{x}$, we derive theorems that lead to an equivalence between a quantum phase-space probability distribution Q(x,p,t) and stochastic trajectories for real amplitudes x and p propagating backwards and forwards in time, respectively. We present forward-backward stochastic simulations that motivate a Q-based model of reality. Amplification plays a key role in measurement. With amplification, contributions due to interference become unobservable, leading to branches that correspond to distinct eigenvalues. This elucidates how the system evolves from a superposition to an eigenstate, from which Born's rule follows. We deduce a hybrid causal structure involving causal deterministic relations for amplified variables, along with microscopic noise inputs and hidden loops for unobservable quantities. Causal consistency is confirmed. The simulations allow evaluation of a state inferred for the system, conditioned on a particular branch, from which we deduce a model for projection and collapse of the wave function. The theory is extended to Einstein-Podolsky-Rosen and Bell nonlocality. We demonstrate consistency with three weak local realistic premises: the existence of real properties (defined after operations that fix measurement settings); a partial locality implying no-signaling; elements of reality that apply to the predictions of a system by a meter, once meter-settings are fixed. A mechanism for non-locality is identified. Our work shows how forward-backward stochastic simulations lead to a hybrid causal structure, involving both deterministic causal relations and hidden stochastic loops, explaining measurement and entanglement, with paradoxes associated with retrocausality avoided.

Figures (35)

• Figure 1: Forward-backward stochastic simulation modeling quantum measurement: Individual trajectories for $x$ and $p$ model a measurement $\hat{x}$ on a system prepared in a superposition $|\psi_{sup}\rangle=\frac{1}{\sqrt{2}}(|x_{1}\rangle+i|-x_{1}\rangle)$ (Eq. (\ref{['eq:sup-sq']}) with $r=2$) of eigenstates of $\hat{x}$. The system is amplified by $H_{amp}$ (Eq. (\ref{['eq:ham-2-1']})), where $t_{0}=0$. The variable $x$ is amplified (left) and $p$ deamplified (right). The trajectories for $x$ propagate in the negative-time direction from a future boundary condition at time $t_{f}=3/g$; those for $p$ propagate forward in time. Two branches for $x$ are evident at time $t_{m}$ when the system is macroscopic. Here, $x_{1}=0.8$, $G=e^{gt}$. (Refer Secs. II - \ref{['sec:Forward-backward-stochastic-simu']}.)
• Figure 2: Causal structure: Top: As for Fig. \ref{['fig:summary-pics']}, with $x_{1}=8$ and $t_{f}=2/g$. Here, the system starts in a superposition of macroscopically distinguishable states. Each branch follows the causal relation $x(0)\rightarrow Gx(0)$ (depicted by the solid blue line). Lower: Diagram of the causal relations applied to the simulation. The Q function $Q(x,p,t_{0})$ describes the system at time $t_{0}$. A future boundary condition is determined by the marginal $Q(x,t_{f})$ in $x$, after amplification for a time $t_{f}$. A deterministic relation (solid two-way blue arrow) connects each $x_{j}$ at time $t_{0}$ to the amplified value $Gx_{j}$ at $t_{f}$. Gaussian noise $\delta x\equiv\eta(t_{f})$ enters at the future boundary $t_{f}$. A connection between forward and backward trajectories $x(t)$ and $p(t)$ at time $t_{0}$ exists, which defines a distribution $Q_{loop}(x,p|x_{j})$, giving rise to a "hidden loop" (orange dashed lines). (Refer main text and Secs. \ref{['sec:Forward-backward-stochastic-simu']} - \ref{['sec:Causal-model-for']} for details).
• Figure 3: Test of causal consistency: How can a future boundary condition be consistent with the causal behavior of the Q function? The diagram illustrates the equivalence of the probability density of the forward- and backward-propagating $p(t)$ and $x(t)$ to the Q function $Q(x,p,t)$, which defines the quantum state $|\psi(t)\rangle$. The system evolves under amplification $H_{amp}$. The equivalence is true for all $t_{f}$, even when $t_{f}$ is far into the future. That the state at time $t$ does not depend on the future inputs is explained by the causal structure of the simulation. An experimental test is feasible. (Refer Sec. \ref{['sec:Causal-model-for']}.)
• Figure 4: Schrodinger's cat paradox: How can cat states be consistent with macroscopic realism? A system is prepared in the superposition $|\psi_{sup}\rangle=\frac{1}{\sqrt{2}}(|x_{1}\rangle+i|-x_{1}\rangle$ of eigenstates $|x_{j}\rangle$ of $\hat{x}$ . The measurement of "which state the cat is in" proceeds by amplification of $\hat{x}$, via $H_{amp}$ (Fig. \ref{['fig:The-causal-structure-1']}). The simulation enables evaluation of $Q_{loop}(x,p|x_{1})$, the postselected distribution inferred for the "cat" at the initial time, given that the final outcome is $x_{1}$. It can be shown that $Q_{loop}(x,p|x_{1})$ is not equivalent to a quantum state $|\psi\rangle$ (refer Sec. \ref{['secQmodel-of']}), highlighting the argument for the incompleteness of quantum mechanics put forward by Schrödinger schrodinger1935gegenwartige. Here, $x_{1}=-x_{2}=1$.
• Figure 5: Causal consistency for Bell-nonlocal systems $A$ and $B$. The Q function $Q(\lambda,t)$ at time $t$ is determined by $|\psi(t)\rangle$ which evolves causally in the positive-time direction, in accordance with Hamiltonians $H_{\theta}^{A}$ and $H_{\phi}^{B}$ that determine the measurement settings, and $H_{amp}^{A}$ and $H_{amp}^{B}$ that amplify $\hat{x}_{\theta A}$ and $\hat{x}_{\phi B}$. Here, $\mathbf{\lambda}=(x_{A},p_{A},x_{B},p_{B})$. The probability density $Q(\lambda,t)$ is equivalent to that of the amplitudes $x_{A}(t)$, $p_{A}(t)$, $x_{B}(t)$, $p_{B}(t)$ (and $x_{\theta K}(t),p_{\theta K}(t)$), as defined by the forward-backward stochastic equations. The $x_{\theta K},p_{\theta K}$ are deterministic functions of $x_{K}$, $p_{K}$ for each $K\in\{A,B\}$ (depicted by two-way crossed blue arrows). The $Q(\lambda,t_{0})$ exhibits local correlation between $x_{K}$ and $p_{K}$ (short vertical black dashed lines), as well as a correlation between $A$ and $B$ (vertical orange dashed line). Branches denoted by $\widetilde{\lambda}_{\theta}^{A}$ and $\widetilde{\lambda}_{\phi}^{B}$ emerge on amplification, where here $\widetilde{\lambda}_{\theta}^{A}$ and $\widetilde{\lambda}_{\phi}^{B}$ are variables that assume the values $x_{\theta j}^{A}$ and $x_{\phi k}^{B}$ representing the outcomes of $\hat{x}_{\theta A}$ and $\hat{x}_{\phi B}$, respectively. The distributions $Q(x_{A},p_{A}|\widetilde{\lambda}_{\phi}^{B})$ and $Q(x_{B},p_{B}|\widetilde{\lambda}_{\theta}^{A})$ determine the probabilities of outcomes at $A$ given the outcome $\widetilde{\lambda}_{\phi}^{B}$ at $B$, and the probabilities of outcomes at $B$ given the outcome $\widetilde{\lambda}_{\theta}^{A}$ at $A$. These are deduced from the backward trajectories, using knowledge of $Q(\lambda,t_{0})$ and deterministic relations, and explain the Bell nonlocality. Being mutually consistent, Lorentz-invariance paradoxes are avoided. (Refer to Sec. IX for details.)