Multiple-shot and unambiguous discrimination of von Neumann measurements

TL;DR

This paper thoroughly investigates the problem of distinguishing von Neumann measurements in finite-dimensional quantum systems, providing analytical results for the minimal number of queries needed for perfect discrimination and demonstrating the effectiveness of parallel strategies.

Contribution

It offers new analytical insights into the minimal queries for perfect discrimination and shows that adaptive strategies do not outperform parallel schemes in this context.

Findings

All pairs of distinct von Neumann measurements can be perfectly distinguished with finite queries.

Optimal discrimination can be achieved with parallel queries, with no advantage from adaptive methods.

Two queries suffice to perfectly distinguish typical Haar-random von Neumann measurement pairs.

Abstract

We present an in-depth study of the problem of multiple-shot discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: minimum error and unambiguous discrimination. In the case of minimum error discrimination, we focus on discrimination of measurements with the assistance of entanglement. We provide an alternative proof of the fact that all pairs of distinct von Neumann measurements can be distinguished perfectly (i.e. with the unit success probability) using only a finite number of queries. Moreover, we analytically find the minimal number of queries needed for perfect discrimination. We also show that in this scenario querying the measurements gives the optimal strategy, and hence any possible adaptive methods do not offer any advantage over the parallel scheme. In the unambiguous discrimination scenario, we give the general expressions for the optimal discrimination probabilities with and without the assistance of entanglement. Finally, we show that typical pairs of Haar-random von Neumann measurements can be perfectly distinguished with only two queries.