The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$) captures computational difficulties of quantum state testing with respect to the trace distance for efficiently preparable mixed states (Quantum State Distinguishability Problem, QSDP), as introduced by Watrous (FOCS 2002). However, this class faces the same parameter issue as its classical counterpart, because of error reduction for the QSDP (the polarization lemma), as demonstrated by Sahai and Vadhan (JACM, 2003). In this paper, we introduce quantum analogues of triangular discrimination, which is a symmetric version of the $\chi^2$ divergence, and investigate the quantum state testing problems for quantum triangular discrimination and quantum Jensen-Shannon divergence (a symmetric version of the quantum relative entropy). These new $\mathsf{QSZK}$-complete problems allow us to improve the parameter regime for testing quantum states in trace distance and examine the limitations of existing approaches to polarization. Additionally, we prove that the quantum state testing for trace distance with negligible errors is in $\mathsf{PP}$ while the same problem without error is in $\mathsf{BQP}_1$. This result suggests that achieving length-preserving polarization for QSDP seems implausible unless $\mathsf{QSZK}$ is in $\mathsf{PP}$.

翻译：复杂性类量子统计零知识（$\mathsf{QSZK}$）刻画了在迹距离下高效可制备混合态（量子态区分问题，QSDP）的量子态测试计算困难性，由Watrous（FOCS 2002）提出。然而，该类别面临与其经典对应类相同的参数问题，原因在于QSDP的误差约化（极化引理），如Sahai和Vadhan（JACM, 2003）所示。本文引入量子三角判别（$\chi^2$散度的对称版本）的量子类比，并研究针对量子三角判别和量子Jensen-Shannon散度（量子相对熵的对称版本）的量子态测试问题。这些新的$\mathsf{QSZK}$完全问题使我们能够改进迹距离下量子态测试的参数体制，并检验现有极化方法的局限性。此外，我们证明具有可忽略误差的迹距离量子态测试属于$\mathsf{PP}$，而无误差的同一问题属于$\mathsf{BQP}_1$。这一结果表明，除非$\mathsf{QSZK}$包含于$\mathsf{PP}$，否则实现QSDP的保长极化似乎不太可能。