Post-measurement states are (very) useful for measurement discrimination

TL;DR

This work introduces Lüders instrument discrimination to quantify the value of post-measurement quantum states in measurement discrimination. For dichotomic qubit projective measurements, access to post-measurement states reduces the problem to discriminating two-copy pure states, yielding a Helstrom-type bound $p_{\text{succ}} = \tfrac{1}{2}\left(1+\sqrt{1-4p_{\psi}p_{\phi}|\langle\psi|\phi\rangle|^4}\right)$, with both entangled and unentangled strategies achieving the optimum. The authors define an instrument advantage bias $\Delta = d_L/d_M$ and prove that it can be arbitrarily large, illustrating potentially huge benefits from post-measurement data even in qubit scenarios (e.g., $\mathcal{Z}$ vs $\mathcal{W}^p$ as $p\to0$). Complementing the analytic results, they perform SDP-based numerical investigations (e.g., trine measurements and $\mathcal{Z}$ vs $\mathcal{W}^p$) to demonstrate operational relevance beyond tractable cases. Overall, the paper highlights post-measurement states as a valuable resource for quantum discrimination tasks and provides both exact and numerical tools to quantify their impact.

Abstract

The standard approach to quantum measurement discrimination is to perform the given unknown measurement on a probe state, possibly entangled with an auxiliary system, and make a decision based on the measurement outcome obtained. In this work, we go beyond the standard aforementioned scenarios by consider not only the classical measurement outcome of a measurement, but also its the post-measurement quantum state. More specifically, instead of considering only the positive-operator valued measure (POVM) operators, we consider their associated Lüders' instrument associated with them. We prove that, when the post-measurement quantum states are available, the task of discriminating two qubit projective measurements is equivalent to discriminating two copies of quantum states associated to each projector pair, extending previous results known for the case where probe states are separable. Then, we proceed by showing that the advantage of considering post-measurement states in measurement discrimination can be large. We formalise this claim by presenting a family of pairs of measurements where the ratio between the discrimination bias of the measurement discrimination task with and without post-measurement states can be arbitrarily large. This shows that, while the post-measurement state was neglected in most of the previous literature, its use can significantly improve the performance of quantum measurement discrimination.

Figures (9)

• Figure 1: The most general strategy for measurement discrimination without the post-measurement state. In yellow: Elements of the strategy (free variables to be optimized over). In red: Unknown objects being discriminated (fixed parameters that define the discrimination problem). $M_{a|x} \in \mathcal{M}_x$ is the POVM element with outcome $a$. $M'_{b|a'} \in \mathcal{M}'_{a'}$ is the POVM element with outcome $b$. $\rho$ is a bipartite input state. For an outcome $a$ of $\mathcal{M}_x$, $f(a)=a'$ where $f$ is a function that decides which measurement $\mathcal{M}'_{a'}$ is to be applied. For an outcome $b$ of $\mathcal{M}'_{a'}$, $g(b)=b'$ decides the final guess of the strategy.
• Figure 2: Most General Strategy for Instrument Discrimination in Circuit Notation. In yellow: Elements of the strategy (Free variables to be optimized over). In red: Unknown objects being discriminated (Fixed parameters that define the discrimination problem). $\widetilde{L}_{a|x} \in \mathcal{L}_x$ is the instrument element with outcome $a$. $M'_{b|a'} \in \mathcal{M}'_{a'}$ is the POVM element with outcome $b$. $\rho$ is a bipartite input state. For an outcome $a$ of $\mathcal{L}_x$, $f(a)=a'$ decides which measurement $\mathcal{M}'_{a'}$ is to be applied. For an outcome $b$ of $\mathcal{M}'_{a'}$, $g(b)=b'$ decides the final guess of the strategy.
• Figure 3: Instrument Discrimination as a Channel Discrimination Task. In yellow: Elements of the strategy (Free variables to be optimized over). In red: Unknown objects being discriminated (Fixed parameters that define the discrimination problem). To the right of the reader, the most general instrument discrimination strategy, the left the equivalent channel discrimination strategy that using testers.
• Figure 4: Success probability of discriminating measurements $\Lambda(0,0)$ and $\Lambda(\theta,\phi)$. On the vertical axis: $\phi$ values in radians. On the horizontal axis: $\theta$ values in radians. The color map represents the success probability with each of the small black separator lines covering a difference of 0.002 .
• Figure 5: Success probability of discriminating Lüders' Instruments Corresponding to $\Lambda(0,0)$ and $\Lambda(\theta,\phi)$. On the vertical axis: $\phi$ values in radians. On the horizontal axis: $\theta$ values in radians. The color map represents the success probability with each of the small black separator lines covering a difference of 0.002 .

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof