Forward-backward simulations of Einstein-Podolsky-Rosen correlations and Bell nonlocality reveal hidden causal loops

A quantum measurement of x is modelled as amplification. We solve for the measurement dynamics in terms of stochastic phase space variables based on the Q function. The solutions lead to amplitudes x and p that propagate backward and forward in the time direction, respectively, with future and past boundary conditions. The trajectories are simulated, the backward-propagating variable requiring a future noise input at the vacuum level. We prove the equivalence between the joint density of amplitudes and the Q function. Causal relations are inferred from the simulations. For superpositions of eigenstates of x, we find that the backward and forward-propagating trajectories are connected by a conditional boundary condition at the initial time, which creates the origin of a causal loop for variables that can be shown to be “hidden” (i.e. not observable). For mixtures, this connection is lost. The simulations reveal a causal structure consistent with macroscopic realism, despite the inherent retrocausality. We simulate Einstein-Podolsky-Rosen entanglement and Bell nonlocality for continuous-variable measurements, revealing similar hidden causal loops. The simulations suggest that a weak form of local realism defined for systems after the measurement settings are fixed is valid, and that Bell nonlocality emerges as a breakdown of a subset of Bell’s local-realism assumptions.