Strategies for optimal single-shot discrimination of quantum measurements

TL;DR

This paper investigates methods for single-shot discrimination of quantum measurements, providing criteria for perfect discrimination, formulas for optimal success probability, and computational tools for measurement distance, with applications to Fourier matrices and mirror isometries.

Contribution

It introduces a comprehensive framework for discriminating von Neumann measurements using both entanglement-free and entangled strategies, including new criteria, formulas, and semidefinite programming approaches.

Findings

Necessary and sufficient conditions for perfect discrimination.

Exact expression for optimal discrimination probability.

Semidefinite and cone programming methods for measurement distance.

Abstract

In this work we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.