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.