最优量子对称性测试中的资源兼容性
Resource Compatibility in Optimal Quantum Symmetry Testing
- The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
- Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）
- QudeLeap Research（QudeLeap研究）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究最优量子对称性测试中资源兼容性问题，通过表示论反向结论揭示零型I错误协议中资源需求与系统边缘态结构约束的关联，并给出多个具体对称群下的最优性与资源条件。
AI中文摘要：
能否在不使用与对称性自然关联的量子资源的情况下，以最优方式进行对称性测试？对于并行子群对Haar测试，我们为每个具有零型I错误的协议推导出一个定量的表示论反向结论：超出的型II错误被两个非负惩罚项之和所下界约束，一个惩罚项对应于非最优子群类型上的权重，另一个惩罚项对应于类型内Schur分布对其类型极值分布的偏离。在最优情况下，这两个惩罚项都消失，从而为每个最优系统边缘态提供支持和权重的锁定。由此产生的锁定集合与自由边缘态集合的不相交性排除了无资源最优性，而仅凭重叠并不足以确立可达性。对于维度$d\geq 2$中的对角环面，无相干性的最优测试仅可能在单次查询时实现。我们确定了在所有具有非相干系统边缘态的协议中的精确最优型II错误，以及达到无限制最优所需的最小系统边缘态相干性。对于维度$d\geq 2$中的正交子群，实协议在每次查询次数下都能达到最优。对于qutrit Clifford群，Wigner正的最优协议在三次和四次查询时存在，但从五次到九次查询时不存在；九次查询之后是否存在仍是一个开放问题。综合来看，这些结果将最优对称性测试中依赖于子群和查询次数的资源需求与最优系统边缘态的结构约束联系起来。
英文摘要：
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 $d\geq 2$, 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 $d\geq 2$, 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.
