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. 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 indicates that the length-preserving polarization for the QSDP implies that $\mathsf{QSZK}$ is in $\mathsf{PP}$.

翻译：复杂性类量子统计零知识（$\mathsf{QSZK}$）刻画了针对可有效制备的混合态（量子态可区分性问题，QSDP）在迹距离下进行量子态检验的计算难度，该概念由Watrous（FOCS 2002）提出。然而，正如Sahai和Vadhan（JACM, 2003）所示，由于QSDP的错误约减（极化引理），该类面临与其经典对应物相同的参数问题。本文引入量子三角判别（$\chi^2$散度的对称版本）的量子类比，并研究量子三角判别与量子Jensen-Shannon散度（量子相对熵的对称版本）的量子态检验问题。这些新的$\mathsf{QSZK}$完全问题使我们能够改进迹距离下量子态检验的参数区间。此外，我们证明，具有可忽略错误的迹距离量子态检验属于$\mathsf{PP}$，而无错误的同一问题则属于$\mathsf{BQP}_1$。这表明QSDP的长度保持极化意味着$\mathsf{QSZK}$包含于$\mathsf{PP}$。