通过一次性量子模糊进行量子资源测试的样本复杂度
Sample complexity of quantum resource testing via one-shot quantum blurring
中文总结 AI 辅助
研究量子资源测试的样本复杂度，通过建立严格有限\(n\)界，给出达到规定性能所需副本数估计，解决了正则化Rényi相对熵收敛问题，得到非对称资源测试样本复杂度界。
中文摘要 AI 辅助
量子资源测试是量子信息处理的基本原语，与资源操纵紧密相连。其目标是从所有自由（即无资源）状态中区分出给定有资源状态\(\rho\)的\(n\)个副本，应用的关键实例是纠缠测试和量子魔术测试。渐近特征依赖于最近证明的广义量子斯坦引理，但其本质上是渐近的，无法提供有限资源保证。本文建立了量子资源测试及量子资源操纵的首个严格有限\(n\)界，给出达到规定性能所需副本数的明确估计。结果包括资源的正则化Rényi相对熵的收敛，解决了重要开放问题；以及非对称资源测试的首个样本复杂度界：对于任何固定的误报错误概率，当\(\delta \to 0\)时，用\(n = O\left(\frac{\log(1/\delta)}{D^\infty(\rho\|F)}\right)\)个\(\rho\)副本可实现至多为\(\delta\)的漏报错误概率。
英文摘要
Quantum resource testing is a fundamental primitive of quantum information processing, profoundly connected to resource manipulation. Its goal is to discriminate $n$ copies of a given resourceful state $ρ$ from all free (i.e., resourceless) states; key instances for applications are entanglement testing and quantum magic testing. The asymptotic characterisation relies on the recently proven Generalised Quantum Stein's Lemma (GQSL), which establishes the rate of decay of the false negative error probability for a fixed false positive error probability. This result, however, is intrinsically asymptotic and thus can provide no finite-resource guarantees, which makes its practical implications unclear. Here, we establish the first rigorous finite-$n$ bounds on quantum resource testing and hence quantum resource manipulation, thus strengthening the GQSL and providing explicit estimates on the number of copies needed to achieve a prescribed performance. As notable consequences, we obtain (a) the convergence of the regularised Rényi relative entropies of a resource, which settles the important open problem from [Fang/Hayashi, arXiv:2508.12901, IEEE ToIT 72:6, 2026]; and (b) the first sample-complexity bound for asymmetric resource testing: for any fixed false positive error probability, a false negative error probability of at most $δ$ can be achieved with $n=O\left(\frac{\log(1/δ)}{D^\infty(ρ\|F)}\right)$ copies of $ρ$, in the limit where $δ\to 0$.
