Information scrambling and chaos induced by a Hermitian matrix

Abstract

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 Poincar'e map} at energy \(E\) and a
corresponding stochastic
\textit{classical Poincar'e-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 Poincar'e map and classical
Poincar'e-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} \(\Lambda(E)\) 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.