Non Classical Causal Correlations via Quasi-Probability Distributions Over Hidden Variables

The observed correlations of the measurement outcomes of an experiment are constrained by the underlying causal structure. For classical causal models with a single latent variable, the observable probability constraints can be derived by implicitizing the probabilities of the hidden transformations. However, quantum mechanics can violate some of these classical constraints; for instance, in the Bell scenario, measurements on an entangled state yield probability outcomes that violate the classical CHSH inequality.

In this project we took a reverse approach: starting with an observable probability distribution, we found the probabilities associated with the deterministic functions of the hidden variable by solving a system of linear equations. Using Hardy’s possibilistic constraints in the Bell scenario, we identified a non maximally entangled quantum state and two simple measurement bases that violate the classical conditions. From the observable probability outcomes, we derived a quasi-probability distribution — characterized by some negative values — for the underlying hidden causal functions.

Our results demonstrate that quantum correlations can, in principle, be modeled by hidden variables frameworks, provided that some of the deterministic functions are permitted to have negative probabilities, thus highlighting the role of quasi-probabilities in capturing non-classical causal relations.