Bell non-locality and Kochen-Specker contextuality: How are they connected?

Abstract

Bell non-locality and Kochen-Specker (KS) contextuality are logically independent concepts, fuel different protocols with quantum vs classical advantage, and have distinct classical simulation costs. A natural question is what are the relations between these concepts, advantages, and costs. To address this question, it is useful to have a map that captures all the connections between Bell non-locality and KS contextuality in quantum theory. The aim of this work is to introduce such a map. After defining the theory-independent notions of Bell non-locality and KS contextuality for ideal measurements, we show that, in quantum theory, due to Neumark's dilation theorem, every matrix of quantum Bell non-local correlations can be mapped to an identical matrix of KS contextual correlations produced in a scenario with identical relations of compatibility but where measurements are ideal and no space-like separation is required. A more difficult problem is identifying connections in the opposite direction. We show that there are "one-to-one" and partial connections between KS contextual correlations and Bell non-local correlations for some KS contextuality scenarios, but not for all of them. However, there is also a method that transforms any matrix of KS contextual correlations for quantum systems of dimension $d$ into a matrix of Bell non-local correlations between two quantum subsystems each of them of dimension $d$. We collect all these connections in map and list some problems which can benefit from this map.

13 pages, 2 figures

References in corpus (14)

• Device-independent security of quantum cryptography against collective attacks
• A simple test for hidden variables in spin-1 system
• Experimentally testable state-independent quantum contextuality
• The Communication Cost of Simulating Bell Correlations
• Wigner function negativity and contextuality in quantum computation on rebits
• Universality of state-independent violation of correlation inequalities for noncontextual theories
• Necessary and sufficient condition for quantum state-independent contextuality
• Converting contextuality into nonlocality
• Proposed experiments of qutrit state-independent contextuality and two-qutrit contextuality-based nonlocality
• Measurement sharpness cuts nonlocality and contextuality in every physical theory
• Bounding quantum contextuality with lack of third-order intereference
• E. Specker: "The logic of non-simultaneously decidable propositions" (1960)
• Bound states emerging from below the continuum in a solvable PT-symmetric discrete Schroedinger equation
• Quantum correlations with a gap between the sequential and spatial cases

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• An Effective Way of Characterizing the Quantum Nonlocality
• Nonclassical correlations of radiation in relation to Bell nonlocality
• Exploring the boundary of quantum correlations with a time-domain optical processor
• Cost of Locally Approximating High-Dimensional Ground States of Contextual Quantum Models
• Twenty years of quantum contextuality at USTC
• Trade-off relations between Bell nonlocality and local Kochen-Specker contextuality in generalized Bell scenarios
• Simultaneous All-versus-Nothing Refutation of Local Realism and Noncontextuality by a Single System