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

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.

7 pages, no figures

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• Algebraic maximum violation of the Leggett-Garg inequality in a two-level system under PT symmetric dynamics
• Formulation of Leggett-Garg inequalities in terms of $q$-entropies
• Protecting temporal correlations of two-qubit states using quantum channels with memory
• Certified Quantumness via Single-Shot Temporal Measurements
• Contextual extensions of quantum gravity models
• Quantumness of electron transport in quantum dot devices through Leggett-Garg inequalities: A non-equilibrium Green's function approach