The robustness of contextuality and the contextuality cost of empirical models

Abstract

In this paper, we introduce and discuss the robustness of contextuality (RoC) R C (e) and the contextuality cost C(e) of an empirical model e. The following properties of them are proved. (i) An empirical model e is contextual if and only if R C (e) > 0; (ii) the RoC function R C is convex, lower semi-continuous and un-increasing under an affine mapping on the set EM of all empirical models; (iii) e is non-contextual if and only if C(e) = 0; (iv) e is contextual if and only if C(e) > 0; (v) e is strongly contextual if and only if C(e) = 1. Also, a relationship between R C (e) and C(e) is obtained. Lastly, the RoC of three empirical models is computed and compared. Especially, the RoC of the PR boxes is obtained and the supremum 0.5 is found for the RoC of all no-signaling type (2, 2, 2) empirical models. relative robustness, robustness of contextuality, contextuality cost, empirical model PACS number(s): 03.67.Mn, 03.65.Ta, 03.65.Ud, 03.65.Db Citation: H. X. Meng, H. X. Cao, and W. H. Wang, The robustness of contextuality and the contextuality cost of empirical models, Sci. China-Phys. Mech.

Generalized Robustness of Contextuality

“….) such that lim n→∞ e n = e. By the closedness of NSEM ( [12], Theorem 2.2), we obtain that e ∈ NSEM.…”

“…Due to the importance of contextuality and motivated by a work on the robustness of entanglement of mixed quantum states against noise and jamming [11], we have proposed and discussed the robustness of contextuality (RoC) R C (e) of an empirical model e in [12,13] to quantify the amount of contextuality with respect to non-contextual mixing by asking about the minimal amount of contextuality-free mixing needed to wipe out all contextuality of e. The mathematical definition is as follows.…”

“…Using the norm-distance of EM in [12], a sequence {e n } ∞ n=1 in EM is convergent to e if and only if e n C i (s) → e C i (s) as n → ∞ for all C i ∈ M, s ∈ E (C i ). We see from [22] (Section 0.4.6) that there exist invertible matrices A and B and identity matrix I such that:…”

“…Let R g (e) = R g (e e ) for some e ∈ EM such that γ e,e (y) ∈ NCEM where y = R g (e e ). For any f ∈ NCEM, we see from the convexity of NCEM [12] that:…”

“…Moreover, contextuality has been formulated in terms of "empirical models", i.e., families of probability distributions [10]. Due to the importance of contextuality and motivated by a work on the robustness of entanglement of mixed quantum states against noise and jamming [11], we proposed and discussed recently in [12,13] the robustness of contextuality (RoC) R C (e) and the contextuality cost C(e) of an empirical model e, where R C (e) denotes the minimal amount of contextuality-free mixing needed to wipe out all contextuality of e and C(e) denotes the minimal amount of contextuality mixing needed to prepare e. The following conclusions have been proven: (i) an empirical model e is contextual if and only if R C (e) > 0; (ii) the robustness of contextuality (RoC) is convex and un-increasing under a non-contextuality-preserving affine mapping; (iii) RoC is bounded and continuous on the set of all no-signaling empirical models; (iv) e is non-contextual if and only if C(e) = 0; and (v) e is strongly contextual if and only if C(e) = 1. Furthermore, a relationship between R C (e) and C(e) has been obtained.…”