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On the distinguishability of geometrically uniform quantum states

Juntai Zhou, Stefano Chessa, Eric Chitambar, Felix Leditzky

TL;DR

The paper develops a comprehensive framework for discriminating geometrically uniform quantum state ensembles, leveraging symmetry to obtain covariant optimal measurements and sharp expressions for success probabilities. It shows that for GU ensembles arising from irreducible representations the optimal success probability is $p_{ ext{succ}}^* = \frac{d}{n}\lambda_{ ext{max}}(\rho)$ with a covariant optimal POVM, and that the $\alpha$-power-PGM converges to optimal as $\alpha\to\infty$ while remaining strictly suboptimal for finite $\alpha$ when the generator is mixed. The authors extend these results to reducible representations via Schur-Weyl duality, and provide explicit analyses for Werner- and permutation-invariant generator states, including the pure-state case where the PGM is optimal with simple success formulas. They apply the framework to concrete tasks such as the hidden subgroup problem over semidirect-product groups and port-based teleportation, deriving simplified, representation-theoretic proofs of optimality and connecting to known sample-complexity results. A general lower bound on PGM in the GU setting yields practical, easily evaluated bounds on success probabilities and sample complexity, enriching the toolkit for quantum information tasks that exhibit high symmetry.

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].

On the distinguishability of geometrically uniform quantum states

TL;DR

The paper develops a comprehensive framework for discriminating geometrically uniform quantum state ensembles, leveraging symmetry to obtain covariant optimal measurements and sharp expressions for success probabilities. It shows that for GU ensembles arising from irreducible representations the optimal success probability is with a covariant optimal POVM, and that the -power-PGM converges to optimal as while remaining strictly suboptimal for finite when the generator is mixed. The authors extend these results to reducible representations via Schur-Weyl duality, and provide explicit analyses for Werner- and permutation-invariant generator states, including the pure-state case where the PGM is optimal with simple success formulas. They apply the framework to concrete tasks such as the hidden subgroup problem over semidirect-product groups and port-based teleportation, deriving simplified, representation-theoretic proofs of optimality and connecting to known sample-complexity results. A general lower bound on PGM in the GU setting yields practical, easily evaluated bounds on success probabilities and sample complexity, enriching the toolkit for quantum information tasks that exhibit high symmetry.

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 . In this work we analyze the problem of discriminating GU ensembles from various angles. Assuming that the representation of 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 -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].
Paper Structure (30 sections, 12 theorems, 105 equations)

This paper contains 30 sections, 12 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Main results and structure of the paper
  3. Preliminaries
  4. Notation
  5. Quantum State Discrimination
  6. Hidden subgroup problem
  7. Port-based teleportation
  8. Geometrically uniform ensembles
  9. Irreducible representations
  10. Success probability and optimal measurement in GU state discrimination
  11. Suboptimality of $\alpha$-PPGM for generic $\alpha$
  12. State exclusion problem
  13. Generator with symmetry
  14. Schur-Weyl Duality
  15. Quantum designs
  16. ...and 15 more sections

Key Result

Proposition 1

Let $\mathcal{H}=\bigoplus_{j=1}^k V_j\otimes W_j$ be a direct-sum decomposition of a Hilbert space $\mathcal{H}$ and let $(p_i,\rho_i)_{i=1}^{N}$ be an ensemble of quantum states on $\mathcal{H}$ of the form for all $i=1,\dots,N$, where $\rho_{ij}$ and $\omega_j$ are quantum states for $j=1,\dots,k$. Then the discrimination task between $(p_i,\rho_i)_{i=1}^{N}$ can be reduced to discriminating b

Theorems & Definitions (27)

  • Proposition 1
  • proof
  • Corollary 2
  • proof
  • Remark 3
  • Example 4: Port-based teleportation
  • Example 5: Dihedral hidden subgroup problem
  • Example 6: Quantum coupon collector problem
  • Lemma 7
  • proof
  • ...and 17 more