Unifying contextual advantages in state discrimination

TL;DR

The paper addresses whether quantum contextuality, framed via Spekkens' generalised contextuality, can universally boost performance in state discrimination tasks. It develops noncontextual inequalities and explicit bounds for MESD, USD, and MCM, showing that quantum implementations can surpass noncontextual limits in both single-shot confidences and average success metrics. Key contributions include the MESD bound $P_g^{(NC)} = 1 - \tfrac{c_{1,2}}{2}$, USD bound $P_0^{(NC)} = \tfrac{1}{2}(1 + c_{1,2})$, and MCM confidence bounds, plus a unifying framework and mirror-ensemble techniques that reveal contextual advantages across all schemes and figures of merit $P_g$, $P_0$, and ${\rm C}(i)$. The results have practical impact for quantum information tasks such as randomness generation and sequential communications, especially in noisy, real-world scenarios where confidences and inconclusive outcomes provide robust witnesses of contextuality.

Abstract

Quantum state discrimination, alongside its other applications, has recently found use as a tool for witnessing generalised contextuality. In this article, we derive noncontextuality inequalities for both conclusive and inconclusive outcomes across various guessing strategies. For minimum- error discrimination, the advantage is in terms of the confidences of individual outcomes, while for unambiguous state discrimination, it is in terms of the average guessing probability. For maximum- confidence discrimination, we show that contextual advantages occur not only for the confidence but also their average, the guessing probability, as well as the inconclusive outcome rate. Our results unify the contextual advantages across all state discrimination schemes and figures of merit. We envisage that various quantum information applications based on state discrimination may offer advantages over non-contextual theories.

Figures (4)

• Figure 1: A general state discrimination scheme requires a part to send one state from an ensemble $\{q_i, \rho_i\}_{i=1}^N$ to another. The latter measures with a POVM which contains both conclusive ($\pi_1,...,\pi_N$) and inconclusive ($\pi_0$) outcomes.
• Figure 2: The confidences found when a minimum error measurement is performed are displayed for both quantum and noncontextual theories. In a quantum measurement both arms have equal confidence $\mathrm{C}^{(Q)}(1)=\mathrm{C}^{(Q)}(2) = \mathrm{C}^{(Q)}(i)$, displayed as a solid black line. In noncontextual theories the two arms have different confidences (blue dashed and red dotted lines) depending on the measurement, parameterised by $\omega$. We use fixed overlap $c_{1,2} = |\langle \psi_1|\psi_2\rangle|=1/2$.
• Figure 3: The inconclusive outcome probability $\mathrm{P}_0$ in both quantum (solid black line) and noncontextual (dashed blue line) theories for the task of discriminating two noisy states characterised by a noise parameter $p$ and overlap $c_{1,2}$ is plotted for varying $p$ (fixed overlap $c_{1,2}=1/2$) and varying $c_{1,2}$ (fixed noise $p=3/4$).
• Figure 4: The guessing probability by quantum (black solid line) and noncontextual (blue dashed line) implementations of a maximum confidence measurement on a binary, noisy ensemble characterised by probability $p=0.5$ as it varies with confusability $c_{1,2}$.