Graph-Theoretic Analysis of $n$-Replica Time Evolution in the Brownian Gaussian Unitary Ensemble

TL;DR

This paper introduces a graph-theoretic method to analyze the $n$-replica time evolution in the Brownian GUE, providing explicit formulas for small $n$ and a general approach for arbitrary $n$, linking disordered systems and quantum information.

Contribution

It develops a novel graph-based framework for calculating $n$-replica time evolution in the BGUE, including explicit cases and a general method for any $n$.

Findings

Explicit formulas for $n=2$ and $n=3$ cases.

A general approach for arbitrary $n$.

Insights into the connection between disordered systems and quantum information.

Abstract

In this paper, we investigate the $n$-replica time evolution operator $\mathcal{U}_n(t)\equiv e^{\mathcal{L}_nt} $ for the Brownian Gaussian Unitary Ensemble (BGUE) using a graph-theoretic approach. We examine the moments of the generating operator $\mathcal{L}_n$, which governs the Euclidean time evolution within an auxiliary $D^{2n}$-dimensional Hilbert space, where $D$ represents the dimension of the Hilbert space for the original system. Explicit representations for the cases of $n = 2$ and $n = 3$ are derived, emphasizing the role of graph categorization in simplifying calculations. Furthermore, we present a general approach to streamline the calculation of time evolution for arbitrary $n$, supported by a detailed example of $n = 4$. Our results demonstrate that the $n$-replica framework not only facilitates the evaluation of various observables but also provides valuable insights into the relationship between Brownian disordered systems and quantum information theory.