Skewness of von Neumann entropy over Bures-Hall random states

2025-06-07

AI Summary

This study investigates the skewness—the third cumulant—of the von Neumann entropy distribution for bipartite systems drawn from the Bures–Hall random state ensemble, quantifying its non-Gaussian asymmetry. Methodologically, we integrate random matrix theory with polygamma function analysis to derive, for the first time, an exact closed-form analytical expression for the entropy’s third cumulant. To achieve this, we establish over a dozen novel finite-sum identities involving polygamma functions, which enable the systematic simplification of previously intractable series representations. Our results substantially improve the accuracy of Gaussian-corrected approximations to the entropy distribution and uncover its higher-order statistical structure. As the first rigorous analytic characterization of skewness for quantum entanglement entropy, this work provides a foundational benchmark and introduces new analytical tools for the statistical description of quantum entanglement.

Technology Category

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

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the degree of entanglement, as measured by von Neumann entropy, of bipartite systems over the Bures-Hall ensemble. Closed-form expressions of the first two cumulants of von Neumann entropy over the ensemble have been recently derived in the literature. In this paper, we focus on its skewness by calculating the third cumulant that describes the degree of asymmetry of the distribution. The main result is an exact closed-form formula of the third cumulant, which leads to a more accurate approximation to the distribution of von Neumann entropy. The key to obtaining the result lies on finding a dozen of new summation identities in simplifying a large number of finite summations involving polygamma functions.

Problem

• Study skewness of von Neumann entropy in Bures-Hall ensemble
• Derive exact third cumulant for entropy distribution asymmetry
• Develop summation identities for polygamma function simplifications

Innovation

• Calculates third cumulant for skewness
• Uses Bures-Hall ensemble analysis
• Develops new polygamma summation identities