Quantum state testing beyond the polarizing regime and quantum triangular discrimination

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 $\mathsf{PP}$.