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Given an arbitrary \(V \times V\) Hermitian matrix, considered as a finitediscrete quantum Hamiltonian, we use methods from graph and ergodic theories toconstruct a corresponding stochastic classical dynamics on an appropriatediscrete phase space. It consists of the directed edges of a graph with \(V\)vertices that are in one-to-one correspondence with the non-vanishingoff-diagonal elements of \(H\). The classical dynamics is a stochastic variantof a Poincar\'e map at an energy \(E\) and an alternative to standardquantum-classical correspondence based on a classical limit \(\hbar \to 0\).Most importantly it can be constructed where no such limit exists. Usingstandard methods from ergodic theory we then proceed to define an expressionfor the Lyapunov exponent \(\Lambda(E)\) of the classical map. It measures therate of separation of stochastic classical trajectories in phase space. Wesuggest to use this Lyapunov exponent to quantify the amount of chaos in afinite quantum system.

How Chaotic is the Dynamics Induced by a Hermitian Matrix?