  We discuss a quantum Kolmogorov-Sinai entropy defined as the entropy
production per unit time resulting from coupling the system to a weak,
auxiliary bath. The expressions we obtain are fully quantum, but require that
the system is such that there is a separation between the Ehrenfest and the
correlation timescales. We show that they reduce to the classical definition in
the semiclassical limit, one instance where this separation holds. We show a
quantum (Pesin) relation between this entropy and the sum of positive
eigenvalues of a matrix describing phase-space expansion. Generalizations to
the case where entropy grows sublinearly with time are possible.
