Entropic Test of Quantum Contextuality

We study the contextuality of a three-level quantum system using classical conditional entropy of measurement outcomes. First, we analytically construct the minimal configuration of measurements required to reveal contextuality. Next, an entropic contextual inequality is formulated, analogous to the entropic Bell inequalities derived by Braunstein and Caves [Phys. Rev. Lett. 61, 662 (1988)], that must be satisfied by all noncontextual theories. We find optimal measurements for violation of this inequality. The approach is easily extendable to higher dimensional quantum systems and more measurements. Our theoretical findings can be verified in the laboratory with current technology.

Figure 1

Graphical notation of commutation relations. Vertices of the graph correspond to observables and edges represent commutativity.

Figure 2

Two examples of graphs of observables admitting joint probability distributions.

Figure 3

Configuration of projectors leading to maximal violation of the entropic contextual inequality for three-level systems.

Figure 4

The plot of positive part of $\mathcal{C}$ as a function of $\theta$ and $\phi$. Maximal violation of the entropic contextual inequality is seen to be 0.091 bits.