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Information scrambling and chaos induced by a Hermitian Matrix

Sven Gnutzmann, Uzy Smilansky

Abstract

Given an arbitrary \(V \times V\) Hermitian matrix, considered as a finite discrete quantum Hamiltonian, we use methods from graph and ergodic theories to construct a \textit{quantum Poincaré map} at energy \(E\) and a corresponding stochastic \textit{classical Poincaré-Markov map} at the same energy on an appropriate discrete \textit{phase space}. This phase space 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 \(\hbar \to 0\). Most importantly it can be constructed where no such limit exists. Using standard methods from ergodic theory we then proceed to define an expression for the \textit{Lyapunov exponent} \(Λ(E)\) of the classical map. It measures the rate of loss of classical information in the dynamics and relates it to the separation of stochastic \textit{classical trajectories} in the phase space. We suggest that loss of information in the underlying classical dynamics is an indicator for quantum information scrambling.

Information scrambling and chaos induced by a Hermitian Matrix

Abstract

Given an arbitrary Hermitian matrix, considered as a finite discrete quantum Hamiltonian, we use methods from graph and ergodic theories to construct a \textit{quantum Poincaré map} at energy and a corresponding stochastic \textit{classical Poincaré-Markov map} at the same energy on an appropriate discrete \textit{phase space}. This phase space consists of the directed edges of a graph with vertices that are in one-to-one correspondence with the non-vanishing off-diagonal elements of . 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 . Most importantly it can be constructed where no such limit exists. Using standard methods from ergodic theory we then proceed to define an expression for the \textit{Lyapunov exponent} \(Λ(E)\) of the classical map. It measures the rate of loss of classical information in the dynamics and relates it to the separation of stochastic \textit{classical trajectories} in the phase space. We suggest that loss of information in the underlying classical dynamics is an indicator for quantum information scrambling.
Paper Structure (4 sections, 26 equations, 3 figures)

This paper contains 4 sections, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The classical dynamics associated with a matrix and the Lyapunov exponent
  3. Instructive examples
  4. Discussion and Outlook

Figures (3)

  • Figure 1: Mean (or, equivalently, local) Lyapunov exponent for $H=A$ on a $d$-regular graph as a function of $E$.
  • Figure 2: (a) Spin graph with $V_{\mathrm{spin}}=4$ vertices. (b) Corresponding graph of the Hamiltonian where the 16 vertices correspond to spin configurations. (c-e) Mean Lyapunov exponent (black), local Lyapunov exponents (coloured lines, colours correspond to the ones used in (b)). The spectrum is located at the black arrows where the height corresponds to the participation ratio divided by $V=16$. We show results for $J_{12}=\frac{1}{3}$, $J_{13}=\frac{\sqrt{5}}{3}$, $J_{23}=\frac{\sqrt{11}}{3}$, and $J_{24}=\frac{1}{\sqrt{3}}$.
  • Figure 3: Squared wavefunctions (Blue, arbitrary scale), local Lyapunov exponents (Red, arbitrary scale), and effective potential $W_{\mathrm{eff}}(n)$ (Black). (a-b) Mean field behaviour for a high (a) and low (b) value of the energy $E$ within the spectrum. (c-d) Single realisation at $\beta=10$ (the dashed line gives the random potential). (e-f) The same for $\beta =0.1$.