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 Rényi 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 $α\in (0, 1)$, the $α$-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 $α$-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.

翻译：量子态之间的相对熵量化了它们的可区分性。已有文献对某些相对熵的估计进行了研究，例如冯·诺依曼相对熵和夹层Rényi相对熵。本文对量子Tsallis相对熵的估计进行了全面研究。我们证明，对于任意常数$α\in (0, 1)$，两个秩为$r$的量子态之间的$α$-Tsallis相对熵可以用样本复杂度$\operatorname{poly}(r)$进行估计；若已知其态制备电路，则可实现更高效的估计。作为应用，我们获得了一种关于量子Hellinger距离的容错量子态认证方法，其样本复杂度为$\widetilde{O}(r^{3.5})$。当$r$是量子比特数的多项式时，该方法相比基于量子态层析的传统方法具有指数级优势。此外，我们证明了在特定参数范围内，关于量子$α$-Tsallis相对熵和量子Hellinger距离的量子态可区分性问题是$\mathsf{QSZK}$-完全的，而在低秩情况下它们是$\mathsf{BQP}$-完全的。