On the distinguishability of geometrically uniform quantum states
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908774631997440 |
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| author | Zhou, Juntai Chessa, Stefano Chitambar, Eric Leditzky, Felix |
| author_facet | Zhou, Juntai Chessa, Stefano Chitambar, Eric Leditzky, Felix |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the distinguishability of geometrically uniform quantum states Zhou, Juntai Chessa, Stefano Chitambar, Eric Leditzky, Felix Quantum Physics 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]. |
| title | On the distinguishability of geometrically uniform quantum states |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2501.12376 |