Contextuality witness inspired by optimal state discrimination

TL;DR

The paper addresses preparation contextuality in quantum theory using a simple prepare-and-measure two-state discrimination scenario. It defines a contextuality witness $W = (p_{suc} - p_{err})/2$ and derives quantum and noncontextual bounds for a fixed inconclusive rate $p_{inc}$, including closed-form expressions for optimal measurements. The quantum bound $W^Q$ is obtained for an ensemble $ ho_x = r_s | ilde{ angle}_x ilde{ angle}_x| + (1-r_s) frac{ ext{I}}{2}$ with overlap $ ext{delta}$, while the noncontextual bound $W^{NC}$ accounts for depolarising noise via $r_s$ and the confusability $c$. The results show that the quantum region can violate the noncontextual bound even with sizeable depolarising noise and losses, and that allowing inconclusive outcomes can enhance robustness, offering a path to contextual advantages in realistic scenarios and suggesting generalisation to multi-state discrimination.

Abstract

Many protocols and tasks in quantum information science rely inherently on the fundamental notion of contextuality to provide advantages over their classical counterparts, and contextuality represents one of the main differences between quantum and classical physics. In this work we present a witness for preparation contextuality inspired by optimal two-state discrimination. The main idea is based on finding the accessible averaged success and error probabilities in both classical and quantum models. We can then construct a noncontextuality inequality and associated witness which we find to be robust against depolarising noise and loss in the form of inconclusive events.

Figures (3)

• Figure 1: Space of probabilities corresponding to a two-state discrimination setting. Continuous lines denote maximum confidence measurements in both quantum (purple) and noncontextual (green) models. Even with a bounded value of noise ($r_{s}=0.7$), the MCM line according to the quantum model falls outside the noncontextual region.
• Figure 2: Bounds on the witness $\mathcal{W}$ according to quantum and noncontextual models. On the first row we show noiseless cases with different overlaps. Below, on the second row, we fix a particular overlap and show the effects of depolarising noise on the preparation. The green area denotes the feasible values according to quantum and noncontextual models and the blue region solely for the quantum model. The black-dashed line shows the contextuality witness $\mathcal{W}^{\ast}$ in Eq. (\ref{['eq:NCineq']}). Any behavior above $\mathcal{W}^{\ast}$ is an evidence of contextuality.
• Figure 3: Tolerable amount of depolarising noise for which our witness can detect quantum contextuality as function of inconclusive rate $p_\text{inc}$ for different overlaps $\delta$. Contextuality is witnessed in the shaded regions above the solid lines (larger $r_s$ means less noise).