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

TL;DR

This paper demonstrates that utilizing post-measurement quantum states can significantly enhance the discrimination of quantum measurements, surpassing traditional methods that rely solely on measurement outcomes.

Contribution

It introduces a framework for measurement discrimination that incorporates post-measurement states, showing their potential to greatly improve discrimination performance.

Findings

Discrimination with post-measurement states is equivalent to discriminating two copies of associated quantum states.

The advantage of using post-measurement states can be arbitrarily large in certain measurement pairs.

Neglecting post-measurement states may lead to suboptimal measurement discrimination strategies.

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 L\"uders' 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.