Quantum Kolmogorov-Sinai entropy and Pesin relation

Consider a pair of classical trajectories starting at two different points in phase-space that are indistinguishable up to some experimental precision. If the two trajectories evolve into distinguishable states after a finite time, we produce information. In classical mechanics, the relation between the Kolmogorov-Sinai (KS) entropy, which measures the rate at which such information is produced, and Lyapunov exponents that characterize how small local variations diverge in time, is well established, and is given by the Pesin formula. In quantum systems, on the other hand, the connection between information spreading and chaos is more subtle. I will present a quantum Kolmogorov-Sinai entropy which is defined as the entropy production per unit time resulting from coupling a system to a weak, auxiliary bath. The derived expressions are fully quantum but require that the system is such that there is a separation between the Ehrenfest and correlation timescales. I will show that our quantum KS entropy reduces to the classical definition in the semiclassical limit, and yields a quantum Pesin-like formula when four-point correlation functions grow exponentially in time, relating the KS entropy to the sum of positive eigenvalues of a matrix that describes phase-space expansion.