Graph-Theoretic Analysis of $n$-Replica Time Evolution in the Brownian Gaussian Unitary Ensemble
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910001343234048 |
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| author | Li, Tingfei Yu, Jianghui |
| author_facet | Li, Tingfei Yu, Jianghui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09681 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graph-Theoretic Analysis of $n$-Replica Time Evolution in the Brownian Gaussian Unitary Ensemble Li, Tingfei Yu, Jianghui Quantum Physics High Energy Physics - Theory 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. |
| title | Graph-Theoretic Analysis of $n$-Replica Time Evolution in the Brownian Gaussian Unitary Ensemble |
| topic | Quantum Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2502.09681 |