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

• URL: http://arxiv.org/abs/2502.09681v1
• Date: Thu, 13 Feb 2025 12:24:50 GMT
• Title: Graph-Theoretic Analysis of $n$-Replica Time Evolution in the Brownian Gaussian Unitary Ensemble
• Authors: Tingfei Li, Cheng Peng, Jianghui Yu,
• Abstract summary: We investigate the $n$-replica time evolution operator $mathcalU_n(t)equiv emathcalL_nt $ for the Brownian Gaussian Unitary Ensemble (BGUE) using a graph-theoretic approach. Explicit representations for the cases of $n = 2$ and $n = 3$ are derived, emphasizing the role of graph categorization in simplifying calculations.
• Score: 3.9404852133765083
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• 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.

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