Axiomatic approach to contextuality and nonlocality

Abstract

We present a unified axiomatic approach to contextuality and non-locality based on the fact that both are resource theories. In those theories the main objects are consistent boxes, which can be transformed by certain operations to achieve certain tasks. The amount of resource is quantified by appropriate measures of the resource. Following recent paper [J.I. de Vicente, J. Phys. A: Math. Theor. {\bf 47}, 424017 (2014)], and recent development of abstract approach to resource theories, such as entanglement theory, we propose axioms and welcome properties for operations and measures of resources. As one of the axioms of the measure we propose the asymptotic continuity: the measure should not differ on boxes that are close to each other by more than the distance with a factor depending logarithmically on the dimension of the boxes. We prove that relative entropy of contextuality is asymptotically continuous. Considering another concept from entanglement theory---the convex roof of a measure---we prove that for some non-local and contextual polytopes, the relative entropy of a resource is upper bounded up to a constant factor by the cost of the resource. Finally, we prove that providing a measure $X$ of resource does not increase under allowed class of operations, such as e.g. wirings, the maximal distillable resource which can be obtained by these operations is bounded from above by the value of $X$ up to a constant factor. We show explicitly which axioms are used in the proofs of presented results, so that analogous results may remain true in other resource theories with analogous axioms. We also make use of the known distillation protocol of bipartite nonlocality to show how contextual resources can be distilled.

17 pages, comments are most welcome

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Cited by in corpus (26)

• Quantum Resource Theories
• Kochen-Specker Contextuality
• Robustness of Measurement, discrimination games and accessible information
• Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes
• Quantifying resource in catalytic resource theory
• The type-independent resource theory of local operations and shared randomness
• Quantum Advantage from Sequential-Transformation Contextuality
• Resource theory of contextuality for arbitrary prepare-and-measure experiments
• Noncontextual wirings
• Nonlocality free wirings and the distinguishability between Bell boxes
• One-Shot Manipulation of Dynamical Quantum Resources
• All states are universal catalysts in quantum thermodynamics
• Quantum information
• The operational advantages provided by non-classical teleportation
• The operational significance of the quantum resource theory of Buscemi nonlocality
• Graph Approach to Extended Contextuality
• Quantum Bell Nonlocality is Entanglement
• Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios
• Anomalies of weight-based coherence measure and mixed maximally coherent states
• Corrected Bell and Noncontextuality Inequalities for Realistic Experiments
• The rank of contextuality
• Limitations on device independent key secure against non signaling adversary via the squashed non-locality
• Optimal allocation of quantum resources
• Algebraic and geometric properties of local transformations
• Convexity of noncontextual wirings and how they order the set of correlations
• Mathematical methods for resource-based type theories