Quantifying Contextuality

Contextuality is central to both the foundations of quantum theory and to the novel information processing tasks. Despite some recent proposals, it still faces a fundamental problem: how to quantify its presence? In this work, we provide a universal framework for quantifying contextuality. We conduct two complementary approaches: (i) the bottom-up approach, where we introduce a communication game, which grasps the phenomenon of contextuality in a quantitative manner; (ii) the top-down approach, where we just postulate two measures, relative entropy of contextuality and contextuality cost, analogous to existent measures of nonlocality (a special case of contextuality). We then match the two approaches by showing that the measure emerging from the communication scenario turns out to be equal to the relative entropy of contextuality. Our framework allows for the quantitative, resource-type comparison of completely different games. We give analytical formulas for the proposed measures for some contextual systems, showing in particular that the Peres-Mermin game is by order of magnitude more contextual than that of Klyachko et al. Furthermore, we explore properties of these measures such as monotonicity or additivity.

Figure 1

Exemplification of contextuality of systems of observables ($A_1,\dots ,A_5$) (a) Contexts (here neighboring $A_i$): observables within each context are jointly measurable so that we can ascribe joint probability within context. (b) Ascribing single common joint probability distribution, which has marginals equal to that ascribed in (a) is not possible. (c) Exemplary possible description of the system: by means of two different common joint probability distributions, neither of which reproduces statistics of some context, the left that of $A_1$, $A_5$ and the right that of $A_3$, $A_4$.

Figure 2

The “which context” game. The adversary (A) creates $A_c$ which has context $c$ as that of a chosen box $B$ such that he minimizes communication from sender (S) to receiver (R).

Figure 3

Values of measure $X_u$ for ${\mathrm{CH}}_{\alpha }^{(n)}$ boxes for $3\le n\le 50$: maximally contextual boxes (upper points, $\alpha =1$); maximally contextual quantum boxes (lower points) with (i) odd $n$, $\alpha =[2\cos (\pi /n)]/[1+\cos (\pi /n)]$ and (ii) even $n$, $\alpha =[1+\cos (\pi /n)]/2$.