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Reverse Bell's Theorem and Relativity of Pre- and Postselection

Guido Bacciagaluppi, Ronnie Hermens

Published Feb 1, 2020• 1 citation
OpenAlex

Abstract

In this paper we prove a Bell's theorem in the setting of postselection ('reverse Bell's theorem'). Specifically, we show under which conditions the Bell inequalities can or cannot be violated with classical postselection, and how this differs from the quantum violations. We then propose a variant of existing experiments that discriminates between quantum violations and classical simulations. The proposed experiment can be adapted to test simultaneously the standard and reverse Bell's theorems. In this case, the distinction between these pre- and postselection effects becomes foliation-dependent.

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Introduction Fifty years after the publication of Bell's theorem, there remains some controversy regarding what the theorem is telling us about quantum mechanics, and what the experimental violations of Bell inequalities are telling us about the world. This chapter represents my best attempt to be clear about what I think the lessons are. In brief: There is some sort of nonlocality inherent in any quantum theory, and, moreover, in any theory that reproduces, even approximately, the quantum probabilities for the outcomes of experiments. But not all forms of nonlocality are the same; there is a distinction to be made between action at a distance and other forms of nonlocality, and I will argue that the nonlocality needed to violate the Bell inequalities need not involve action at a distance. Furthermore, the distinction between forms of nonlocality makes a difference when it comes to compatibility with relativistic causal structure. The Bell locality condition is a condition which, if satisfied, renders possible a completely local account of the correlations between outcomes of spatially separated experiments. Bell's theorem tells us that the probabilities that we compute from quantum mechanics do not admit of such an account. Modulo auxiliary assumptions that, in my opinion, should be noncontroversial, this yields the conclusion that there must be something nonlocal about any theory that yields the quantum mechanical probabilities or anything close to them. One way to violate the Bell locality condition – and this is the way that any deterministic theory must do it – is via straightforward action at a distance. However, we should also consider the possibility of a stochastic theorywith probabilistic laws that involve irreducible correlations. The sort of nonlocality inherent in such a theory involves relations between events that are significantly different from causal relations as usually conceived. Perhaps surprisingly, such a theory, unlike its deterministic cousins, need not invoke a distinguished relation of distant simultaneity. This makes a difference for the situation of the two sorts of theories vis-a-vis relativity. Though a deterministic theory can achieve the appearance of being relativistic, at the level of observable phenomena, it can do so only by introducing a preferred foliation and with it a causal structure at odds with relativistic causal structures. Things are otherwise with dynamical collapse theories.

Cambridge University Press eBooks•Aug 31, 2016•15 citations
In this paper we prove a Bell's theorem in the setting of postselection ('reverse Bell's theorem'). Specifically, we show under which conditions the Bell inequalities can or cannot be violated with classical postselection, and how this differs from the quantum violations. We then propose a variant of existing experiments that discriminates between quantum violations and classical simulations. The proposed experiment can be adapted to test simultaneously the standard and reverse Bell's theorems. In this case, the distinction between these pre- and postselection effects becomes foliation-dependent.
The Bell inequalities can be violated by postselecting on the results of a measurement of the Bell states. If information about the original state preparation is available, we point out how the violation can be reproduced classically by postselecting on the basis of this information. We thus propose a variant of existing experiments that rules out such alternative explanations by having the preparation and the postselection at spacelike separation. Unlike the timelike case in which one can sharply distinguish Bell inequality violations based on pre- or postselection of a Bell state, in our scenario the distinction between these physical effects becomes foliation dependent. We call this ``relativity of pre- and postselection'' and conclude from it that quantum state evolution is not a fundamental process and that we should adopt an event-based Heisenbergian picture instead.
Recently Wang et al. have reported a violation of a Bell inequality without entanglement [arXiv:2507.07756]. We show that their result is an artifact of postselection. It is well known that postselection may yield Bell inequality violations, both in classical toy models and in real experiments with delayed-choice entanglement-swapping. Here we describe a classical analog of Wang et al.'s experiment, and show that it produces essentially the same results as their quantum version. We explain in detail why neither version is a challenge to Local Causality or local realism: the postselection entails a rejection of Bell's assumption of Statistical Independence.
Bell's theorem shows that local realistic theories place strong restrictions on observable correlations between different systems, giving rise to Bell's inequality which can be violated in experiments using entangled quantum states. Bell's theorem is based on the assumptions of realism, locality, and the freedom to choose between measurement settings. In experimental tests, "loopholes" arise which allow observed violations to still be explained by local realistic theories. Violating Bell's inequality while simultaneously closing all such loopholes is one of the most significant still open challenges in fundamental physics today. In this paper, we present an experiment that violates Bell's inequality while simultaneously closing the locality loophole and addressing the freedom-of-choice loophole, also closing the latter within a reasonable set of assumptions. We also explain that the locality and freedom-of-choice loopholes can be closed only within nondeterminism, i.e., in the context of stochastic local realism.
In classical theory, the trajectory of a particle is entirely predetermined by the complete set of initial conditions via dynamical laws. Based on this, we formulate a no-go theorem for the dynamics of classical particles, i.e., a Bell inequality for trajectories, and discuss its possible violation in a quantum scenario. A trajectory, however, is not an outcome of a quantum measurement, in the sense that there is no observable associated with it, and thus there is no ``direct'' experimental test of the Bell inequality for trajectories. Nevertheless, we show how to overcome this problem by considering a special case of our generic inequality that can be experimentally tested point-by-point in time. Such inequality is indeed violated by quantum mechanics, and the violation persists during an entire interval of time and not just at a particular singular instant. We interpret the violation to imply that trajectories (or at least pieces thereof) cannot exist predetermined, within a local-realistic theory.
Introduction Fifty years after the publication of Bell's theorem, there remains some controversy regarding what the theorem is telling us about quantum mechanics, and what the experimental violations of Bell inequalities are telling us about the world. This chapter represents my best attempt to be clear about what I think the lessons are. In brief: There is some sort of nonlocality inherent in any quantum theory, and, moreover, in any theory that reproduces, even approximately, the quantum probabilities for the outcomes of experiments. But not all forms of nonlocality are the same; there is a distinction to be made between action at a distance and other forms of nonlocality, and I will argue that the nonlocality needed to violate the Bell inequalities need not involve action at a distance. Furthermore, the distinction between forms of nonlocality makes a difference when it comes to compatibility with relativistic causal structure. The Bell locality condition is a condition which, if satisfied, renders possible a completely local account of the correlations between outcomes of spatially separated experiments. Bell's theorem tells us that the probabilities that we compute from quantum mechanics do not admit of such an account. Modulo auxiliary assumptions that, in my opinion, should be noncontroversial, this yields the conclusion that there must be something nonlocal about any theory that yields the quantum mechanical probabilities or anything close to them. One way to violate the Bell locality condition – and this is the way that any deterministic theory must do it – is via straightforward action at a distance. However, we should also consider the possibility of a stochastic theorywith probabilistic laws that involve irreducible correlations. The sort of nonlocality inherent in such a theory involves relations between events that are significantly different from causal relations as usually conceived. Perhaps surprisingly, such a theory, unlike its deterministic cousins, need not invoke a distinguished relation of distant simultaneity. This makes a difference for the situation of the two sorts of theories vis-a-vis relativity. Though a deterministic theory can achieve the appearance of being relativistic, at the level of observable phenomena, it can do so only by introducing a preferred foliation and with it a causal structure at odds with relativistic causal structures. Things are otherwise with dynamical collapse theories.