Resource Compatibility in Optimal Quantum Symmetry Testing
Abstract
Can a symmetry be tested optimally without using the quantum resource naturally associated with it? For parallel subgroup-versus-Haar testing, we derive a quantitative representation-theoretic converse for every protocol with zero type-I error: the excess type-II error is bounded below by the sum of two nonnegative penalties, one for weight on suboptimal subgroup types and one for deviations of within-type Schur profiles from their typewise extremal profiles. At optimality, both penalties vanish, yielding support and weight locking for every optimal system marginal. Disjointness of the resulting locked set and the free marginal set rules out resource-free optimality, whereas overlap alone does not establish attainability. For the diagonal torus in dimension d≥ 2, optimal testing without coherence is possible only with a single query. We determine both the exact optimal type-II error over all protocols with incoherent system marginals and the minimum system-marginal coherence required to attain the unrestricted optimum. For the orthogonal subgroup in dimension d≥ 2, a real protocol attains the optimum at every query number. For the qutrit Clifford group, Wigner-positive optimal protocols exist at three and four queries but not from five through nine; whether they exist beyond nine queries remains open. Together, these results relate the subgroup- and query-dependent resource requirements of optimal symmetry testing to structural constraints on optimal system marginals.
Related
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BibTeX
@misc{cao2026resource,
title = {{Resource Compatibility in Optimal Quantum Symmetry Testing}},
author = {Yimeng Cao and Ranyiliu Chen and Chengkai Zhu and Xin Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38134},
eprint = {2609.38134},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38134},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Resource Compatibility in Optimal Quantum Symmetry Testing
AU - Yimeng Cao
AU - Ranyiliu Chen
AU - Chengkai Zhu
AU - Xin Wang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38134
UR - https://arxiv.org/abs/2609.38134
AB - Can a symmetry be tested optimally without using the quantum resource naturally associated with it? For parallel subgroup-versus-Haar testing, we derive a quantitative representation-theoretic converse for every protocol with zero type-I error: the excess type-II error is bounded below by the sum of two nonnegative penalties, one for weight on suboptimal subgroup types and one for deviations of within-type Schur profiles from their typewise extremal profiles. At optimality, both penalties vanish, yielding support and weight locking for every optimal system marginal. Disjointness of the resulting locked set and the free marginal set rules out resource-free optimality, whereas overlap alone does not establish attainability. For the diagonal torus in dimension d≥ 2, optimal testing without coherence is possible only with a single query. We determine both the exact optimal type-II error over all protocols with incoherent system marginals and the minimum system-marginal coherence required to attain the unrestricted optimum. For the orthogonal subgroup in dimension d≥ 2, a real protocol attains the optimum at every query number. For the qutrit Clifford group, Wigner-positive optimal protocols exist at three and four queries but not from five through nine; whether they exist beyond nine queries remains open. Together, these results relate the subgroup- and query-dependent resource requirements of optimal symmetry testing to structural constraints on optimal system marginals.
ER -