Layer Cake Representations for Quantum Divergences

2025-07-09

AI Summary

The construction of quantum divergences has long suffered from insufficient mathematical rigor and intuitive clarity due to operator noncommutativity. Method: This work introduces the “layer-cake representation” into quantum divergence theory—integrating quantum information theory, noncommutative integration, operator monotone function analysis, and layer-cake techniques—to establish a unified, intuitive, and rigorous framework for quantum Rényi divergences and f-divergences. Contributions/Results: We provide a novel proof of the integral representation of quantum relative entropy; confirm a long-standing conjecture on the trace formula for Rényi divergences; and propose two new variational representations—namely, an optimization-based variational form and a Riemann–Stieltjes-type operator integral representation. Rigorous equivalence between these representations and existing integral formulations is established. The framework is validated in the error exponent analysis of binary quantum hypothesis testing. These results deepen the fundamental understanding of quantum relative entropy and Rényi divergences and broaden their mathematical foundations in information-theoretic applications.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The resulting quantum Rényi and $f$-divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relative entropy by Frenkel and prove a conjecture regarding a trace expression for the Rényi divergence. Additionally, we give applications to error exponents in hypothesis testing, a new Riemann-Stieltjes type integral representation and a variational representation.

Problem

• Extending classical divergences to quantum information challenges
• Proposing layer cake representation for quantum divergences
• Proving equivalences and applications in hypothesis testing

Innovation

• Layer cake representation for quantum divergences
• Proving equivalence with integral representations
• Applications in hypothesis testing error exponents