On the distinguishability of geometrically uniform quantum states

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].