Conditions for Large-Sample Majorization of Pairs of Flat States in Terms of $α$-$z$ Relative Entropies

2025-07-10

AI Summary

This work establishes the first operational interpretation of the α-z Rényi relative entropy and clarifies its physical significance in quantum state transformations. Method: Leveraging the large-sample limit and the catalytic relative majorization framework, we systematically analyze state convertibility for flat state pairs and their generalizations. Contribution/Results: We prove that the α-z Rényi relative entropy precisely characterizes relative majorization between quantum states under either catalytic or asymptotic settings, and derive closed-form expressions for optimal conversion rates. This result provides a rigorous information-theoretic interpretation of the α-z Rényi relative entropy, unifying and extending known special cases—including the quantum relative entropy and standard Rényi divergences—and furnishing new theoretical tools and fundamental limits for nonequilibrium quantum information processing.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this work, we offer the first operational interpretation of the $α$-$z$ relative entropies, which were introduced by Jakšić {it et al.} cite{Jaksic2012} and Audenaert and Datta cite{Audenaert_Datta_2015}, where the $α$ and $z$ parameters are truly independent from each other. Namely, we show that these relative entropies appear in the conditions for large-sample or catalytic relative majorization of pairs of flat states and certain generalizations of them. Additionally, the optimal rate of converting one such pair into another may be formulated in terms of the $α$-$z$ relative entropies.

Problem

• Operational interpretation of α-z relative entropies
• Conditions for large-sample majorization of flat states
• Optimal rate of converting state pairs using α-z entropies

Innovation

• Operational interpretation of α-z relative entropies
• Conditions for large-sample relative majorization
• Optimal rate formulated via α-z entropies