Generalized Robustness of Contextuality

Motivated by the importance of contextuality and a work on the robustness of the entanglement of mixed quantum states, the robustness of contextuality (RoC) RC(e) of an empirical model e against non-contextual noises was introduced and discussed in Science China Physics, Mechanics and Astronomy (59(4) and 59(9), 2016). Because noises are not always non-contextual, this paper introduces and discusses the generalized robustness of contextuality (GRoC) Rg(e) of an empirical model e against general noises. It is proven that Rg(e) = 0 if and only if e is non-contextual. This means that the quantity Rg can be used to distinguish contextual empirical models from non-contextual ones. It is also shown that the function Rg is convex on the set of all empirical models and continuous on the set of all no-signaling empirical models. For any two empirical models e and f such that the generalized relative robustness of e with respect to f is finite, a fascinating relationship between the GRoCs of e and f is proven, which reads Rg(e)Rg( f ) ≤ 1. Lastly, for any n-cycle contextual box e, a relationship between the GRoC Rg(e) and the extent ∆e of violating the non-contextual inequalities is established.