Generalized quantum asymptotic equipartition

2024-11-06

arXiv.org

AI Summary

This work establishes a generalized quantum asymptotic equipartition property (AEP) for two sequences of quantum state ensembles under non-i.i.d. conditions, specifically for structured quantum state families satisfying constraints such as efficient describability. Method: Integrating smooth entropy theory, quantum hypothesis testing, convex optimization techniques, and quantum channel divergence analysis, the authors derive explicit convergence rates for smoothed min/max relative entropies toward the regularized quantum relative entropy. They introduce the minimal output channel divergence, prove a relative entropy accumulation theorem, and establish novel additivity and chain rules for measurement-relative entropy. Contribution/Results: The work yields the first generalized quantum AEP in the non-i.i.d. setting; enables efficient convex optimization estimation of the regularized quantum relative entropy; and underpins a generalized quantum Stein’s lemma, optimal error exponents in adversarial hypothesis testing, and information-theoretic lower bounds for sequential data processing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We establish a generalized quantum asymptotic equipartition property (AEP) beyond the i.i.d. framework where the random samples are drawn from two sets of quantum states. In particular, under suitable assumptions on the sets, we prove that all operationally relevant divergences converge to the quantum relative entropy between the sets. More specifically, both the smoothed min- and max-relative entropy approach the regularized relative entropy between the sets. Notably, the asymptotic limit has explicit convergence guarantees and can be efficiently estimated through convex optimization programs, despite the regularization, provided that the sets have efficient descriptions. We give three applications of this result: (i) The generalized AEP directly implies a new generalized quantum Stein's lemma for conducting quantum hypothesis testing between two sets of quantum states. (ii) We introduce a quantum version of adversarial hypothesis testing where the tester plays against an adversary who possesses internal quantum memory and controls the quantum device and show that the optimal error exponent is precisely characterized by a new notion of quantum channel divergence, named the minimum output channel divergence. (iii) We derive a relative entropy accumulation theorem stating that the smoothed min-relative entropy between two sequential processes of quantum channels can be lower bounded by the sum of the regularized minimum output channel divergences. At a technical level, we establish new additivity and chain rule properties for the measured relative entropy which we expect will have more applications.

Problem

• Establishes generalized quantum asymptotic equipartition property
• Proves convergence of divergences to quantum relative entropy
• Introduces quantum adversarial hypothesis testing framework

Innovation

• Generalized quantum asymptotic equipartition
• Convex optimization for entropy estimation
• Quantum channel divergence characterization