turkmath.org

Given an arbitrary Hermitian matrix H (considered as a quantum system) we use methods from graph and ergodic theories to construct a quantum Poincaré map (a sparse unitary map that contains full information on the spectrum) and a corresponding stochastic classical Poincaré-Markov map on an appropriate discrete phase space. This phase space consists of the directed edges of a graph 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. 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 mean Lyapunov exponent 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. The talk is based on a joint work with Uzy Smilansky.