Unification of Bell, Leggett-Garg and Kochen-Specker inequalities: A hybrid spatio-temporal inequality

TL;DR

This paper unifies Bell, Leggett-Garg, and Kochen-Specker inequalities into a single framework, revealing deep connections between spatial, temporal, and contextual quantum correlations, and introduces new hybrid inequalities with potential cryptographic applications.

Contribution

It develops a unified correlation inequality framework and derives novel hybrid spatio-temporal inequalities extending previous methods.

Findings

Quantum mechanics violates all the unified inequalities.

The framework links local, contextual, and temporal correlations.

Implications for cryptography and monogamy relations are discussed.

Abstract

The Bell-type (spatial), Kochen-Specker (contextuality) or Leggett-Garg (temporal) inequalities are based on classically plausible but otherwise quite distinct assumptions. For any of these inequalities, satisfaction is equivalent to a joint probability distribution for all observables in the experiment. This implies a joint distribution for all pairs of observables, and is indifferent to whether or not they commute in the theory. This indifference underpins a unification of the above inequalities into a general framework of correlation inequalities. When the physical scenario is such that the correlated pairs are all compatible, the resulting correlation is non-signaling, which may be local or multi-particle, corresponding to contextuality or Bell-type inequalities. If the pairs are incompatible, the resulting correlation corresponds to Leggett-Garg (LG) inequalities. That quantum mechanics (QM) violates all these inequalities suggests a close connection between the local, spatial and temporal properties of the theory. As a concrete manifestation of the unification, we extend the method due to Roy and Singh (J. Phys. A, 11, L167 (1978)) to derive and study a new class of hybrid spatio-temporal inequalities, where the correlated pairs in the experiment are both compatible or incompatible. The implications for cryptography and monogamy inequalities of the unification are briefly touched upon.