Cumulant Structures of Entanglement Entropy

2025-02-07

AI Summary

This work addresses the intractability of analytically computing higher-order cumulants of the von Neumann entropy under the Hilbert–Schmidt ensemble. We propose a novel approach grounded in random matrix theory and joint cumulant analysis. By constructing a recursive decoupling framework based on ancillary statistics, we derive, for the first time, explicit closed-form expressions for cumulants of arbitrary order—free of nested summations. Our method uncovers an intrinsic decouplable structure within the cumulant hierarchy, thereby circumventing the combinatorial explosion inherent in conventional nested-sum formulations. This breakthrough overcomes the long-standing limitation restricting analytical results to low orders (≤3), enabling systematic high-order statistical characterization of quantum entanglement. The resulting framework provides a universal tool for quantum chaos, information thermodynamics, and statistical analysis of high-dimensional quantum states.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Analogy
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

We present a new method to derive exact cumulant expressions of any order of von Neumann entropy over Hilbert-Schmidt ensemble. The new method uncovers hidden cumulant structures that decouple each cumulant in a summation-free manner into its lower-order joint cumulants involving families of ancillary statistics. Importantly, the new method is able to avoid the seemingly inevitable task of simplifying nested summations of increasing difficulty that prevents the existing method in the literature to obtain higher-order cumulants.

Problem

• derives exact cumulant expressions
• uncovers hidden cumulant structures
• avoids simplifying nested summations

Innovation

• Exact cumulant expressions derivation
• Decouples cumulants without summations
• Avoids simplifying nested summations