Information scrambling and chaos induced by a Hermitian Matrix

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.