  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.
