Generalizing Bell-type and Leggett-Garg-type Inequalities to Systems with Signaling

Abstract

Contextuality means non-existence of a joint distribution for random variables recorded under mutually incompatible conditions, subject to certain constraints imposed on how the identity of these variables may change across these conditions. In simple quantum systems contextuality is indicated by violations of Bell-type or Leggett-Garg-type inequalities. These inequalities, however, are predicated on the assumption of no-signaling, defined as invariance of the distributions of measurement results with respect to other (e.g., earlier in time) measurements' settings. Signaling makes the inequalities inapplicable: a non-signaling system with any degree of contextuality, however high, loses any relation to this concept as soon as it exhibits any degree of signaling, however small. This is unsatisfactory. We describe a principled way of defining and measuring contextuality in arbitrary systems with random outputs, whether signaling is absent or present.

arXiv comment5 pp., 1 fig. In vv. 4 and 5 minor typos are corrected

References in corpus (6)

• General properties of Nonsignaling Theories
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• A Kochen-Specker inequality
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• Order-distance and other metric-like functions on jointly distributed random variables
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Cited by in corpus (10)

• Necessary and Sufficient Conditions for Extended Noncontextuality in a Broad Class of Quantum Mechanical Systems
• Contextuality is About Identity of Random Variables
• Contextuality in Three Types of Quantum-Mechanical Systems
• Leggett-Garg Inequalities, Pilot Waves and Contextuality
• Quantum- vs. Macro- Realism: What does the Leggett-Garg Inequality actually test?
• Negative probabilities and Counterfactual Reasoning on the double-slit Experiment
• Measuring Observable Quantum Contextuality
• From the problem of Future Contingents to Peres-Mermin square experiments: An introductory review to Contextuality
• Testing Contextuality in Cyclic Psychophysical Systems of High Ranks
• Indistinguishability and Negative Probabilities