{"title":"Information scrambling and chaos induced by a Hermitian matrix","authors":"Sven Gnutzmann, Uzy Smilansky","doi":"10.1088/1751-8121/ad6f7c","DOIUrl":null,"url":null,"abstract":"Given an arbitrary <italic toggle=\"yes\">V</italic> × <italic toggle=\"yes\">V</italic> Hermitian matrix <italic toggle=\"yes\">H</italic>, considered as a finite discrete quantum Hamiltonian, we use methods from graph and ergodic theories to construct a <italic toggle=\"yes\">quantum Poincaré map</italic> at energy <italic toggle=\"yes\">E</italic> and a corresponding stochastic <italic toggle=\"yes\">classical Poincaré–Markov map</italic> at the same energy on an appropriate discrete <italic toggle=\"yes\">phase space</italic>. This phase space <inline-formula>\n<tex-math><?CDATA $\\mathcal{D}$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"aad6f7cieqn1.gif\"></inline-graphic></inline-formula> consists of the directed edges of a graph with <italic toggle=\"yes\">V</italic> vertices that are in one-to-one correspondence with the non-vanishing off-diagonal elements of <italic toggle=\"yes\">H</italic>. The correspondence between quantum Poincaré map and classical Poincaré–Markov map is an alternative to the standard quantum–classical correspondence based on a classical limit <inline-formula>\n<tex-math><?CDATA $\\hbar \\to 0$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>ℏ</mml:mi><mml:mo accent=\"false\" stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"aad6f7cieqn2.gif\"></inline-graphic></inline-formula>. Most importantly it can be constructed where no such limit exists. Using standard methods from ergodic theory we then proceed to derive an expression for the <italic toggle=\"yes\">Lyapunov exponent</italic> <inline-formula>\n<tex-math><?CDATA $\\Lambda(E)$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"normal\">Λ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"aad6f7cieqn3.gif\"></inline-graphic></inline-formula> of the classical map. It measures the rate of loss of classical information in the dynamics and relates it to the separation of stochastic <italic toggle=\"yes\">classical trajectories</italic> in the phase space. We suggest that loss of information in the underlying classical dynamics is an indicator for quantum information scrambling.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":null,"pages":null},"PeriodicalIF":2.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Physics A: Mathematical and Theoretical","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1751-8121/ad6f7c","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
Given an arbitrary V × V Hermitian matrix H, considered as a finite discrete quantum Hamiltonian, we use methods from graph and ergodic theories to construct a quantum Poincaré map at energy E and a corresponding stochastic classical Poincaré–Markov map at the same energy on an appropriate discrete phase space. This phase space D consists of the directed edges of a graph with V vertices that are in one-to-one correspondence with the non-vanishing off-diagonal elements of H. The correspondence between quantum Poincaré map and classical Poincaré–Markov map is an alternative to the standard quantum–classical correspondence based on a classical limit ℏ→0. Most importantly it can be constructed where no such limit exists. Using standard methods from ergodic theory we then proceed to derive an expression for the Lyapunov exponentΛ(E) of the classical map. It measures the rate of loss of classical information in the dynamics and relates it to the separation of stochastic classical trajectories in the phase space. We suggest that loss of information in the underlying classical dynamics is an indicator for quantum information scrambling.
给定一个任意的 V × V 赫米矩阵 H,将其视为有限离散量子哈密顿,我们利用图论和遍历理论的方法,在适当的离散相空间上构建能量 E 的量子波卡列图和相同能量的相应随机经典波卡列-马尔科夫图。这个相空间 D 由一个有 V 个顶点的图的有向边组成,这些顶点与 H 的非消失对角元素一一对应。量子波因卡雷图与经典波因卡雷-马尔科夫图之间的对应关系是基于经典极限ℏ→0 的标准量子-经典对应关系的另一种选择。最重要的是,它可以在不存在这种极限的地方构建。利用遍历理论的标准方法,我们可以推导出经典映射的莱普诺夫指数Λ(E) 的表达式。它衡量了动力学中经典信息的损失率,并将其与相空间中随机经典轨迹的分离联系起来。我们认为,底层经典动力学中的信息损失是量子信息扰乱的一个指标。
期刊介绍:
Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.