[Paper Review] Wigner's inequality for conditional probabilities and nonexistence of a realistic model for the two dimensional Hilbert space

Andrei Khrennikov|ArXiv.org|Aug 14, 2003

Quantum Mechanics and Applications18 references3 citations

TL;DR

This paper re-examines Bell's inequality by showing that the classical probabilistic model Bell implicitly used—lacking conditional probabilities defined by Bayes' formula—differs fundamentally from Kolmogorov’s axiomatic probability theory. The key contribution is proving a new Wigner-type inequality for conditional probabilities, which is violated by quantum mechanics even in two-dimensional Hilbert space, demonstrating the nonexistence of a realistic local model for quantum systems at the most basic level.

ABSTRACT

We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be defined by classical Bayes' formula. We also use the approach based on Bell-type inequalities in the conventional probabilistic approach, Kolmogorov model. We prove an analog of Wigner's inequality for conditional probabilities and by using this inequality show that predictions of the conventional and quantum probability models disagree already in the case of noncomposite systems (even in the two dimensional case).

Motivation & Objective

• To clarify the implicit probabilistic model Bell used in his EPR-Bell analysis, which differs from Kolmogorov’s axiomatic probability framework.
• To identify that Bell’s model lacks a proper definition of conditional probability via Bayes’ formula, a core feature of classical probability theory.
• To demonstrate that the disagreement between classical and quantum probability models emerges even in noncomposite systems, such as two-dimensional Hilbert space.
• To establish a new Bell-type inequality based on conditional probabilities, applicable to sequential (non-simultaneous) measurements.
• To show that models like those of Pitowsky and Aerts, which reproduce quantum statistics, necessarily violate this new inequality due to non-Bayesian conditional probabilities.

Proposed method

• The paper constructs a 'Bellian' probability model that uses measure-theoretic foundations but omits conditional probabilities and Bayes’ formula, contrasting it with the standard Kolmogorov model.
• It derives a new Wigner-type inequality for conditional probabilities under the assumption that Bayes’ formula holds, forming a Bell-type constraint on classical probabilistic models.
• The inequality is applied to the two-dimensional Hilbert space, showing that quantum predictions violate it even without entanglement or composite systems.
• The analysis compares classical conditional probabilities (defined via Bayes’ formula) with quantum conditional probabilities (defined via the von Neumann projection postulate).
• It uses the von Neumann correspondence framework to link classical and quantum statistical models, emphasizing the need to include a conditional probability postulate (CP) to close the gap between models.
• The paper analyzes existing hidden-variable models (Pitowsky, Aerts, Czachor) and shows they avoid Bell’s inequality not by consistency with classical probability, but by violating the Bayesian structure of conditional probability.

Experimental results

Research questions

• RQ1 Why does Bell’s original derivation of nonlocality fail to clearly define the probabilistic framework he used, and how does this ambiguity affect interpretations of his result?
• RQ2 Can a Bell-type inequality be derived for conditional probabilities rather than joint probabilities, and what does it imply for classical realism?
• RQ3 Does the violation of such a conditional probability inequality occur even in a two-dimensional Hilbert space, indicating nonlocality or nonrealism at the most fundamental level?
• RQ4 Why do models like those of Pitowsky and Aerts reproduce quantum statistics without violating Bell’s inequality, and what structural feature allows this?
• RQ5 Is the failure of Bayes’ formula in quantum systems the root cause of the incompatibility between classical and quantum probability models?

Key findings

• A new Wigner-type inequality is derived for conditional probabilities under the assumption of Bayes’ formula, which must be satisfied by any classical probabilistic model.
• This inequality is violated by quantum mechanics even in the two-dimensional Hilbert space, proving that no realistic local model can reproduce quantum predictions at the level of conditional probabilities.
• The Bellian probabilistic model used implicitly by Bell does not define conditional probabilities via Bayes’ formula, making it incompatible with standard Kolmogorovian probability theory.
• The violation of the conditional probability inequality arises not from entanglement but from the non-Bayesian structure of quantum conditional probabilities, even in single-particle systems.
• Models by Pitowsky and Aerts avoid Bell’s inequality not by being classical, but because they inherently violate the Bayesian structure of conditional probability, aligning with quantum non-Kolmogorovian behavior.
• The paper concludes that the postulate of conditional probability correspondence (CP) is essential for any classical-quantum correspondence and that its absence renders such models physically incoherent.