The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$) captures computational difficulties of the time-bounded quantum state testing problem with respect to the trace distance, known as the Quantum State Distinguishability Problem (QSDP) introduced by Watrous (FOCS 2002). However, QSDP is in $\mathsf{QSZK}$ merely within the constant polarizing regime, similar to its classical counterpart shown by Sahai and Vadhan (JACM 2003) due to the polarization lemma (error reduction for SDP). Recently, Berman, Degwekar, Rothblum, and Vasudevan (TCC 2019) extended the $\mathsf{SZK}$ containment for SDP beyond the polarizing regime via the time-bounded distribution testing problems with respect to the triangular discrimination and the Jensen-Shannon divergence. Our work introduces proper quantum analogs for these problems by defining quantum counterparts for triangular discrimination. We investigate whether the quantum analogs behave similarly to their classical counterparts and examine the limitations of existing approaches to polarization regarding quantum distances. These new $\mathsf{QSZK}$-complete problems improve $\mathsf{QSZK}$ containments for QSDP beyond the polarizing regime and establish a simple $\mathsf{QSZK}$-hardness for the quantum entropy difference problem (QEDP) defined by Ben-Aroya, Schwartz, and Ta-Shma (ToC 2010). Furthermore, we prove that QSDP with some exponentially small errors is in $\mathsf{PP}$, while the same problem without error is in $\mathsf{NQP}$.

翻译：复杂度类量子统计零知识（$\mathsf{QSZK}$）刻画了关于迹距离的时间有界量子态测试问题的计算困难性，即由Watrous（FOCS 2002）提出的量子态可区分性问题（QSDP）。然而，类似于Sahai和Vadhan（JACM 2003）基于极化引理（SDP的误差缩减）所示的经典对应情形，QSDP仅在恒定极化区间内属于$\mathsf{QSZK}$。近期，Berman、Degwekar、Rothblum和Vasudevan（TCC 2019）通过考虑关于三角判别和詹森-香农散度的时间有界分布测试问题，将SDP的$\mathsf{SZK}$包含关系扩展至极化区间之外。本文通过定义三角判别的量子对应形式，引入了这些问题的恰当量子类似物。我们探究量子类似物是否具有与经典对应物相似的行为，并考察现有方法对量子距离极化问题的局限性。这些新的$\mathsf{QSZK}$完全问题改进了QSDP在极化区间之外的$\mathsf{QSZK}$包含关系，并为Ben-Aroya、Schwartz和Ta-Shma（ToC 2010）定义的量子熵差问题（QEDP）建立了简单的$\mathsf{QSZK}$困难性。此外，我们证明了具有某些指数级小误差的QSDP属于$\mathsf{PP}$，而无误差的同一问题属于$\mathsf{NQP}$。