Pretty simple bounds on quantum state discrimination

TL;DR

This paper demonstrates that the pretty good measurement can efficiently identify an unknown quantum state from a set of mixed states with bounded pairwise fidelities using logarithmic copies, and provides a strategy for pure states with fewer copies.

Contribution

It introduces bounds on quantum state discrimination using the pretty good measurement for mixed and pure states, improving understanding of measurement efficiency.

Findings

Pretty good measurement achieves high success with O(log n) copies for mixed states.

Explicit measurement strategy for pure states with  copies based on Gram matrix.

Provides worst-case guarantees for quantum state discrimination.

Abstract

We show that the quantum measurement known as the pretty good measurement can be used to identify an unknown quantum state picked from any set of $n$ mixed states that have pairwise fidelities upper-bounded by a constant below 1, given $O(\log n)$ copies of the unknown state, with high success probability in the worst case. If the unknown state is promised to be pure, there is an explicit measurement strategy which solves this worst-case quantum state discrimination problem with $\widetilde{O}(\|G\|)$ copies, where $G$ is the Gram matrix of the states.