On Estimating the Quantum Tsallis Relative Entropy

2025-10-01

AI Summary

This work investigates the estimability of the quantum Tsallis relative entropy for α ∈ (0,1). To address this, we propose the first systematic estimation framework for rank-r quantum states, integrating quantum state sampling, extraction of circuit information for state preparation, quantum Hellinger distance analysis, and complexity-theoretic reduction. We prove that the α-Tsallis relative entropy can be estimated efficiently with Õ(r^{3.5}) sample complexity, enabling fault-tolerant quantum state verification—an exponential improvement over conventional quantum tomography. Furthermore, we establish its learnability in the low-rank regime and uncover a deep connection between α-Tsallis relative entropy and QSZK/BQP-complete problems. These results provide novel theoretical tools and foundations for efficient verification and distinguishability analysis of low-rank quantum states, as well as for characterizing quantum computational complexity.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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 $αin (0, 1)$, 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.

Problem

• Estimating quantum Tsallis relative entropy between quantum states efficiently
• Developing tolerant quantum state certification using Hellinger distance
• Characterizing quantum state distinguishability complexity classes

Innovation

• Estimates quantum Tsallis entropy with poly(r) complexity
• Uses state-preparation circuits for enhanced efficiency
• Enables tolerant certification via Hellinger distance optimization