On the distinguishability of geometrically uniform quantum states

TL;DR

This paper investigates the discrimination of geometrically uniform quantum states, establishing optimal measurement strategies and success probabilities, with applications to quantum algorithms like hidden subgroup problems and port-based teleportation.

Contribution

It provides a comprehensive analysis of GU ensemble discrimination, demonstrating the optimality of the pretty good measurement in various scenarios and simplifying proofs for key quantum information tasks.

Findings

Optimal measurement as limit of weighted PGM for irreducible representations

Simplified proofs of PGM optimality in hidden subgroup and teleportation tasks

Lower bounds on success probability for n-copy GU ensembles

Abstract

A geometrically uniform (GU) ensemble is a uniformly weighted quantum state ensemble generated from a fixed state by a unitary representation of a finite group $G$. In this work we analyze the problem of discriminating GU ensembles from various angles. Assuming that the representation of $G$ is irreducible, we first show that a particular optimal measurement can be understood as the limit of weighted `pretty good measurements' (PGM). This naturally provides examples of state discrimination for which the unweighted PGM is provably sub-optimal. We extend this analysis to certain reducible representations, and use Schur-Weyl duality to discuss two particular examples of GU ensembles in terms of Werner-type and permutation-invariant generator states. For the case of pure-state GU ensembles we give a streamlined proof of optimality of the PGM first proved in [Eldar et al., 2004]. We use this result to give simplified proofs of the optimality of the PGM, along with expressions for the corresponding success probabilities, for two tasks: the hidden subgroup problem over semidirect product groups (first proved in [Bacon et al., 2005]), and port-based teleportation (first proved in [Mozrzymas et al., 2019] and [Leditzky, 2022]). Finally, we consider the $n$-copy setting and adapt a result of [Montanaro, 2007] to derive a compact and easily evaluated lower bound on the success probability of the PGM for this task. This result can be applied to the hidden subgroup problem to obtain a new proof for an upper bound on the sample complexity by [Hayashi et al., 2006].