Strategies for single-shot discrimination of process matrices

TL;DR

This paper develops methods for single-shot discrimination of process matrices, providing exact probabilities, optimization techniques, and insights into strategies for quantum combs, advancing understanding of quantum causal structures.

Contribution

It introduces an exact expression for discrimination probability, a semidefinite programming approach, and analyzes strategies for quantum combs, highlighting new tools and results in quantum causality discrimination.

Findings

Exact expression for optimal discrimination probability

Semidefinite programming for process matrix distance

Equal success probability for adaptive and non-signalling strategies with quantum combs

Abstract

The topic of causality has recently gained traction quantum information research. This work examines the problem of single-shot discrimination between process matrices which are an universal method defining a causal structure. We provide an exact expression for the optimal probability of correct distinction. In addition, we present an alternative way to achieve this expression by using the convex cone structure theory. We also express the discrimination task as semidefinite programming. Due to that, we have created the SDP calculating the distance between process matrices and we quantify it in terms of the trace norm. As a valuable by-product, the program finds an optimal realization of the discrimination task. We also find two classes of process matrices which can be distinguished perfectly. Our main result, however, is a consideration of the discrimination task for process matrices corresponding to quantum combs. We study which strategy, adaptive or non-signalling, should be used during the discrimination task. We proved that no matter which strategy you choose, the probability of distinguishing two process matrices being a quantum comb is the same.