[Paper Review] The whole is greater than the sum of the parts: on the possibility of purely statistical interpretations of quantum theory

Joseph Emerson, Dmitry Serbin+2 more|arXiv (Cornell University)|Dec 4, 2013

Quantum Mechanics and Applications17 citations

TL;DR

This paper challenges the Pusey-Barrett-Rudolph (PBR) theorem by showing that its assumption of preparation independence implicitly encodes local causality, not purely statistical interpretations. By introducing a weaker 'local independence' principle that respects no-signaling and is empirically indistinguishable from PBR's assumption when non-local variables are inaccessible, the authors construct an explicit non-local hidden variable model that reproduces quantum predictions, demonstrating that purely statistical interpretations remain viable if local causality is rejected.

ABSTRACT

The Pusey-Barrett-Rudolph theorem (PBR) claims to rule out the possibility of a purely statistical interpretation of the quantum state under an assumption of how to represent independent operations in any hidden variable model. We show that PBR's assumption of independence encodes an assumption of local causality, which is already known to conflict with the predictions of quantum theory via Bell-type inequalities. We devise a weaker formulation of independence within a general hidden variable model that is empirically indistinguishable from the PBR assumption in situations where certain hidden variables are inaccessible. Under this weaker principle we are able to construct an explicit hidden variable model that is purely statistical and also reproduces the quantum predictions. Our results suggest that the assumption of a purely statistical interpretation is actually an innocent bystander in the PBR argument, rather than the driving force behind their contradiction.

Motivation & Objective

• To challenge the conclusion of the PBR theorem that purely statistical interpretations of quantum states are impossible.
• To identify that the PBR assumption of preparation independence is equivalent to an assumption of local causality.
• To propose a weaker notion of independence—'local independence'—that is consistent with no-signaling and empirically indistinguishable from PBR’s assumption under local measurements.
• To construct an explicit non-local hidden variable model that reproduces quantum predictions under this weaker independence condition.
• To argue that the real conflict in the PBR argument lies in local causality, not in the statistical interpretation of quantum states.

Proposed method

• Formalize the PBR scenario using ontological models with preparation measures μψ(λ) and response functions ξk(λ) for measurements.
• Identify that PBR’s preparation independence assumption implies local causality by restricting support of measurement functions to locally accessible variables only.
• Introduce 'local independence' as a weaker alternative that allows non-local hidden variables but preserves no-signaling under local measurements.
• Construct a toy model with non-local variables (e.g., a shared bit) and define preparation distributions μ00, μ0+, μ+0, μ++ over joint ontic states λ1×λ2.
• Show that under local independence, the marginal distributions satisfy preparation independence and reproduce quantum statistics.
• Demonstrate that the PBR contradiction vanishes when non-local variables are inaccessible to local measurements, preserving quantum predictions.

Experimental results

Research questions

• RQ1 Does the PBR theorem genuinely rule out purely statistical interpretations of quantum states, or is the contradiction driven by a different assumption?
• RQ2 Is the PBR assumption of preparation independence equivalent to an assumption of local causality in the context of hidden variable models?
• RQ3 Can a non-local hidden variable model with purely statistical quantum states reproduce quantum predictions under a weaker independence condition?
• RQ4 Is the assumption of preparation independence empirically distinguishable from local independence when non-local variables are inaccessible to local measurements?
• RQ5 Does the failure of local causality—rather than the statistical nature of quantum states—underlie the PBR result?

Key findings

• The PBR theorem’s conclusion that purely statistical interpretations are impossible is invalid because the core conflict arises from the assumption of local causality, not from the statistical interpretation itself.
• The PBR assumption of preparation independence is shown to be equivalent to local causality, a condition already known to be incompatible with quantum mechanics via Bell’s theorem.
• A weaker notion of independence—'local independence'—is introduced, which is consistent with no-signaling and empirically indistinguishable from PBR’s assumption when non-local variables are inaccessible.
• An explicit non-local hidden variable model is constructed that satisfies local independence and reproduces the quantum predictions for the PBR scenario.
• The model shows that purely statistical interpretations of quantum states remain viable if local causality is rejected, suggesting that the statistical interpretation is an 'innocent bystander' in the PBR argument.
• The analysis reveals a duality between the EPR-Bell scenario (entangled states, independent measurements) and the PBR scenario (independent preparations, entangled measurements), both of which are undermined by local causality.