A slightly improved upper bound for quantum statistical zero-knowledge

The complexity class Quantum Statistical Zero-Knowledge $mathsf{QSZK}$, introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound mathsf{QIP(2)} cap co-mathsf{QIP(2)}, which was simplified following the inclusion mathsf{QIP(2)} subseteq mathsf{PSPACE} established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, mathsf{QIP(2)} denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically computationally unbounded, and co-mathsf{QIP(2)} denotes the complement of mathsf{QIP(2)}. We slightly improve this upper bound to mathsf{QIP(2)} cap co-mathsf{QIP(2)} with a quantum linear-space honest prover. A similar improvement also applies to the upper bound for the non-interactive variant mathsf{NIQSZK}. Our main techniques are an algorithmic version of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).