[Paper Review] Classical description of quantum randomness using stochastic gauge systems

M. Feldmann|arXiv (Cornell University)|Nov 8, 2009

Quantum Mechanics and Applications72 references3 citations

TL;DR

This paper proposes a classical probability model called stochastic gauge systems that reproduces quantum randomness and entanglement without non-locality. By making the Kolmogorov probability space context-dependent—dependent on experimental settings—it classically emulates EPR pairs, PR-boxes, GHZ and W states, and derives entanglement entropy via Kullback-Leibler divergence, offering a local realistic alternative to quantum non-locality.

ABSTRACT

We present a classical probability model appropriate to the description of quantum randomness. This tool, that we have called stochastic gauge system, constitutes a contextual scheme in which the Kolmogorov probability space depends upon the experimental setup, in accordance with quantum mechanics. Therefore, the probability space behaves like a gauge parameter. We discuss the technical issues of this theory and apply the concept to classically emulate quantum entangled states and even `super-quantum' systems. We exhibit bipartite examples leading to maximum violation of Bell-CHSH inequalities like EPR pairs or exceeding the Tsirelson bound like PR-boxes, as well as tripartite cases simulating GHZ or W-states. We address also the question of partially correlated systems and multipartite entanglements. In this model, the classical equivalent of the entanglement entropy is identified with the Kullback-Leibler divergence. Hence, we propose a natural generalisation of this function to multipartite systems, leading to a simple evaluation of the degree of entanglement and determining the bounds of maximum entanglement. Finally, we obtain a constructive necessary and sufficient condition of multipartite entanglement.

Motivation & Objective

• To resolve the foundational tension between quantum mechanics and local realism by constructing a classical model that reproduces quantum phenomena without invoking non-locality.
• To address the flaw in Bell’s theorem by showing that absolute probability spaces are physically inconsistent, and instead propose a contextual, gauge-dependent probability framework.
• To provide a classical interpretation of quantum entanglement and entanglement entropy using Kullback-Leibler divergence and boundary-based collapse mechanisms.
• To extend the model to multipartite systems, including GHZ and W states, and to define a constructive criterion for multipartite entanglement.
• To challenge the notion that quantum entanglement is exclusively quantum by showing it can be classically emulated through contextual probability and gauge-invariant randomness.

Proposed method

• Introduces stochastic gauge systems as a contextual probability model where the Kolmogorov probability space depends on the experimental setup, mimicking quantum contextuality.
• Defines 'ignition states' and 'projection functions' to model the classical analog of quantum measurement collapse, triggered by boundary conditions.
• Uses gauge probability distributions to ensure local consistency and contextuality, avoiding the assumption of a universal probability space.
• Applies the Tsirelson criterion to generalize the CHSH inequality violation to more than two regions, enabling simulation of super-quantum systems like PR-boxes.
• Models multipartite entanglement through one-step or multi-step collapse mechanisms, with entanglement entropy identified as Kullback-Leibler divergence between marginal and joint distributions.
• Derives a constructive necessary and sufficient condition for multipartite entanglement based on the violation of local consistency constraints in the gauge framework.

Experimental results

Research questions

• RQ1 Can quantum randomness and entanglement be classically emulated without invoking non-locality?
• RQ2 Does the assumption of an absolute probability space in Bell’s theorem lead to a fundamental flaw, and can this be corrected via a contextual gauge model?
• RQ3 Can classical systems achieve maximum violation of Bell-CHSH inequalities, including super-quantum systems like PR-boxes, within a consistent probability framework?
• RQ4 How can entanglement entropy in quantum systems be classically interpreted and generalized to multipartite systems?
• RQ5 What is the classical analog of quantum collapse, and can it be modeled as a break of equilibrium in an extended system?

Key findings

• The model successfully classically emulates EPR-type entangled states and achieves maximum violation of CHSH inequalities, matching quantum predictions.
• It reproduces PR-box behavior, including super-quantum correlations exceeding the Tsirelson bound, within a consistent classical probability framework.
• The classical equivalent of entanglement entropy is identified as the Kullback-Leibler divergence between joint and marginal probability distributions.
• A constructive necessary and sufficient condition for multipartite entanglement is derived based on the violation of local consistency in the gauge probability model.
• Tripartite systems such as GHZ and W states are successfully simulated using the multi-step collapse mechanism, demonstrating classical analogs of genuine multipartite entanglement.
• The theory provides a classical interpretation of quantum parallelism as the set of all potential outcomes accessible after a classical break of equilibrium, analogous to quantum measurement.