On Estimating the Quantum Tsallis Relative Entropy

The relative entropy between quantum states quantifies their distinguishability. The estimation of certain relative entropies has been investigated in the literature, e.g., the von Neumann relative entropy and sandwiched R\'enyi relative entropy. In this paper, we present a comprehensive study of the estimation of the quantum Tsallis relative entropy. We show that for any constant $\alpha \in (0, 1)$, the $\alpha$-Tsallis relative entropy between two quantum states of rank $r$ can be estimated with sample complexity $\operatorname{poly}(r)$, which can be made more efficient if we know their state-preparation circuits. As an application, we obtain an approach to tolerant quantum state certification with respect to the quantum Hellinger distance with sample complexity $\widetilde{O}(r^{3.5})$, which exponentially outperforms the folklore approach based on quantum state tomography when $r$ is polynomial in the number of qubits. In addition, we show that the quantum state distinguishability problems with respect to the quantum $\alpha$-Tsallis relative entropy and quantum Hellinger distance are $\mathsf{QSZK}$-complete in a certain regime, and they are $\mathsf{BQP}$-complete in the low-rank case.