Quantifying Contextuality

TL;DR

This paper introduces a comprehensive framework for quantifying quantum contextuality using both a communication game approach and information-theoretic measures, establishing their equivalence and analyzing their properties.

Contribution

It proposes a novel quantitative framework for contextuality, linking game-based and information-theoretic measures, with analytical formulas and property analysis.

Findings

The communication game measure equals the relative entropy of contextuality.

Analytical formulas for measures on specific systems are provided.

Properties like monotonicity and additivity are explored.

Abstract

Contextuality is central to both the foundations of quantum theory and to the novel information processing tasks. Although it was recognized before Bell's nonlocality, despite some recent proposals, it still faces a fundamental problem: how to quantify its presence? In this work, we provide a framework for quantifying contextuality. We conduct two complementary approaches: (i) bottom-up approach, where we introduce a communication game, which grasps the phenomenon of contextuality in a quantitative manner; (ii) top-down approach, where we just postulate two measures - relative entropy of contextuality and contextuality cost, analogous to existent measures of non-locality (a special case of contextuality). We then match the two approaches, by showing that the measure emerging from communication scenario turns out to be equal to the relative entropy of contextuality. We give analytical formulas for the proposed measures for some contextual systems. Furthermore we explore properties of these measures such as monotonicity or additivity.