The most discriminable quantum states in the multicopy regime

TL;DR

This paper identifies the most discriminable quantum states in the multi-copy regime, revealing conditions under which pure or mixed states optimize success probability and highlighting quantum advantages over classical systems.

Contribution

It establishes universal limits for multi-copy quantum state discrimination, links classical and quantum problems, and introduces computational methods for optimal ensemble identification.

Findings

Pure state ensembles support maximally discriminable sets when forming a $k$-design.

Mixed states can outperform pure states when the number of states exceeds the $k$-design size.

Quantum systems have a quadratic advantage over classical ones in multi-copy discrimination.

Abstract

This work investigates which sets of quantum states give rise to the highest achievable success probability in minimum-error state discrimination if multiple copies of the unknown state are given. Specifically, we consider uniformly distributed ensembles of the form $\left\{\frac{1}{N},\rho_i^{\otimes k}\right\}_{i=1}^N$, where $N$ states in dimension $d$ are provided in $k$ identical copies, and derive universal limits in this scenario. For pure state ensembles, we prove that whenever $N$ is large enough to support a state $k$-design, these designs will exactly give rise to the maximally discriminable sets. We further show that when $N$ exceeds the size required for a $k$-design, mixed states can outperform all pure state ensembles. We also analyse the analogue classical discrimination problems, in which states are replaced by probability distributions. We recognise that the problem of most discriminable classical states in the multi-copy regime is in one-to-one correspondence to the concept of the multiplicative Bayes capacity of independent uses of classical channels, a concept that emerges naturally in the context of classical information leakage. This connection allows us to completely solve the classical analogue of our problem when $N\geq \binom{d + k - 1}{k}$, and to prove that quantum systems offer a quadratic advantage (in number of copies $k$) over classical ones. Curiously, we also show that this quantum advantage is strongly reduced when one is restricted to real quantum states. Finally, we introduce computational techniques to find sets of most discriminable ensembles, and to obtain rigorous universal upper bounds on the maximal success probability for multi-copy state discrimination in cases that are analytically intractable.