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 $\mathit{\text{quantum}}\mathit{\text{Poincar}}\acute{\mathit{\text{e}}}\mathit{\text{map}}$ at energy $E$ and a corresponding stochastic $\mathit{\text{classical}}\mathit{\text{Poincar}}\acute{\mathit{\text{e}}}\mathit{\text{-Markov}}\mathit{\text{map}}$ at the same energy on an appropriate discrete $\mathit{\text{phase}}\mathit{\text{space}}$. 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 $\mathrm{\hslash }\to 0$. 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 $\mathit{\text{Lyapunov}}\mathit{\text{exponent}}$ $\mathrm{\Lambda }(E)$ of the classical map. It measures the rate of loss of classical information in the dynamics and relates it to the separation of stochastic $\mathit{\text{classical}}\mathit{\text{trajectories}}$ in the phase space. We suggest that loss of information in the underlying classical dynamics is an indicator for quantum information scrambling.