Cumulant Structures of Entanglement Entropy

TL;DR

The paper introduces a summation-free cumulant-structure framework for entanglement entropy S under the Hilbert-Schmidt ensemble, enabling exact κ_l(S) for any order by decoupling each cumulant into lower-order joint cumulants of ancillary statistics. This is achieved through a matrix-level and kernel-level approach centered on a Christoffel-Darboux kernel and new decoupling statistics, which recasts κ_l into recursion with δ_l(k) terms and derivative contributions. The method reproduces known κ_2(S), κ_3(S), and κ_4(S) with concise proofs and, notably, yields new higher-order cumulants κ_5(S) and κ_6(S) in closed form, expressed via higher-order polygamma functions. The framework generalizes beyond the Hilbert-Schmidt setting and promises rapid computation for higher orders and application to other ensembles, advancing the understanding of entanglement fluctuations and tail behavior.

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.

Theorems & Definitions (18)

• Lemma 1
• Lemma 2
• Proposition 1
• Lemma 3
• Proposition 2
• Proposition 3
• Corollary 1
• Theorem 1
• Corollary 2
• proof