[Paper Review] An Operational Interpretation of Negative Probabilities and No-Signalling Models

Samson Abramsky, Adam Brandenburger|arXiv (Cornell University)|Jan 11, 2014

Quantum Mechanics and Applications11 references4 citations

TL;DR

This paper provides an operational interpretation of negative probabilities in quantum foundations by introducing a no-signalling model that simulates contextuality using classical resources with negative probabilities. It demonstrates that such models can reproduce quantum correlations without signaling, offering a classical framework to understand non-classical phenomena in quantum mechanics.

ABSTRACT

Negative probabilities have long been discussed in connection with the foundations of quantum mechanics. We have recently shown that, if signed measures are allowed on the hidden variables, the class of probability models which can be captured by local hidden-variable models are exactly the no-signalling models. However, the question remains of how negative probabilities are to be interpreted. In this paper, we present an operational interpretation of negative probabilities as arising from standard probabilities on signed events. This leads, by virtue of our previous result, to a systematic scheme for simulating arbitrary no-signalling models.

Motivation & Objective

• To provide a physical, operational interpretation of negative probabilities in quantum theory.
• To explore how no-signalling models can simulate quantum contextuality using classical resources with negative probabilities.
• To clarify the role of negative probabilities in non-local correlations and contextuality without signaling.

Proposed method

• Constructing a no-signalling model that uses negative probabilities to simulate quantum correlations.
• Defining a classical framework where outcomes are assigned probabilities, including negative values, consistent with no-signalling constraints.
• Using a bipartite scenario with two parties (Alice and Bob) to model measurement settings and outcomes.
• Introducing a source that prepares a state with negative probabilities, represented as a distribution over outcomes.
• Employing dashed lines and symbolic labels (e.g., 0110, a↦0, b↦1) to represent measurement choices and outcomes in a diagrammatic model.
• Ensuring the model respects no-signalling by verifying that marginal probabilities are independent of distant settings.

Experimental results

Research questions

• RQ1 Can negative probabilities be operationally interpreted in a way that preserves no-signalling in bipartite systems?
• RQ2 How do negative probabilities in a classical model reproduce quantum-like correlations without signaling?
• RQ3 What is the role of contextuality in models with negative probabilities, and how does it relate to quantum non-locality?
• RQ4 Can a classical model with negative probabilities simulate quantum contextuality while respecting no-signalling?
• RQ5 What are the structural and operational constraints on models using negative probabilities in quantum foundations?

Key findings

• Negative probabilities in the model are operationally meaningful and do not lead to signaling, preserving relativistic causality.
• The model successfully reproduces quantum correlations, such as those in the PR-box scenario, using classical resources with negative probabilities.
• The diagrammatic representation shows that outcomes like 0110 are assigned to measurement settings, with consistent marginal probabilities across parties.
• The use of dashed lines and directional arrows indicates that measurement choices are independent, ensuring no signaling between parties.
• The model demonstrates that contextuality can be simulated classically using negative probabilities without violating no-signalling.
• The framework provides a consistent classical interpretation of negative probabilities in quantum foundations, offering a new perspective on quantum contextuality.