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International Journal of Theoretical Physics

Quantifying Contextuality of Empirical Models in Terms of Trace-Distance

Hui-Xian Meng, Huaixin Cao, Wenhua Wang, Yajing Fan, Liang Chen

Published Mar 13, 2017• 1 citation
Publisher
DOI
10.1007/s10773-017-3327-5

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Minimum-error state discrimination between two mixed states $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ can be aided by the receipt of ``classical side information'' specifying which states from some convex decompositions of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ apply in each run. We quantify this phenomena by the average trace distance and give lower and upper bounds on this quantity as functions of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$. The lower bound is simply the trace distance between $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, trivially seen to be tight. The upper bound is $\sqrt{1\ensuremath{-}\mathrm{tr}(\ensuremath{\rho}\ensuremath{\sigma})}$, and we conjecture that this is also tight. We reformulate this conjecture in terms of the existence of a pair of ``unbiased decompositions,'' which may be of independent interest, and prove it for a few special cases. Finally, we point towards a link with a notion of nonclassicality known as preparation contextuality.
Contextuality is a feature of quantum correlations. It is crucial from a foundational perspective as a non-classical phenomenon, and from an applied perspective as a resource for quantum advantage. It is commonly defined in terms of hidden variables, for which it forces a contradiction with the assumptions of parameter-independence and determinism. The former can be justified by the empirical property of non-signalling or non-disturbance, and the latter by the empirical property of measurement sharpness. However, in realistic experiments neither empirical property holds exactly, which leads to possible objections to contextuality as a form of non-classicality, and potential vulnerabilities for supposed quantum advantages. We introduce measures to quantify both properties, and introduce quantified relaxations of the corresponding assumptions. We prove the continuity of a known measure of contextuality, the contextual fraction, which ensures its robustness to noise. We then bound the extent to which these relaxations can account for contextuality, via corrections terms to the contextual fraction (or to any non-contextuality inequality), culminating in a notion of genuine contextuality, which is robust to experimental imperfections. We then show that our result is general enough to apply or relate to a variety of established results and experimental set-ups. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.