On Estimating the Quantum Tsallis Relative Entropy
Jinge Bao, Minbo Gao, Qisheng Wang
TL;DR
This work provides a comprehensive complexity-theoretic and algorithmic treatment of estimating the quantum Tsallis relative entropy ${D}_{\textnormal{Tsa},\alpha}(\rho\|\sigma)$ for constant $\alpha\in(0,1)$ between rank-$r$ quantum states. It develops quantum algorithms achieving poly$(r)$ sample complexity and sublinear query complexity (varying by $\alpha$), leveraging block-encodings, quantum singular value transformation, Hadamard tests, amplitude estimation, and a quantum samplizer to convert queries into samples; this yields efficient tolerant testing for the quantum Hellinger distance with significant gains over tomography in the low-rank regime. The paper also establishes tight computational hardness results, showing $\mathsf{QSZK}$-completeness in general and $\mathsf{BQP}$-completeness in the low-rank regime for Tsallis/Hellinger testers, thereby charting a full picture of the estimation landscape for quantum divergences beyond von Neumann and Rényi entropies. These results enable tomography-free certification and testing of quantum states with provable efficiency guarantees, and open avenues for extending the framework to related divergences and imaginarity-like quantities.
Abstract
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.