The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate given by the sum of all positive Lyapunov exponents of the system. We prove a quantum version of this result valid for bosonic systems with unstable quadratic Hamiltonian. The derivation takes into account the case of time-dependent Hamiltonians with Floquet instabilities. We show that the entanglement entropy of a Gaussian state grows linearly for large times in unstable systems, with a rate determined by the Lyapunov exponents and the choice of the subsystem . We apply our results to the analysis of entanglement production in unstable quadratic potentials and due to periodic quantum quenches in many-body quantum systems. Our results are relevant for quantum field theory, for which we present three applications: a scalar field in a symmetry-breaking potential, parametric resonance during post-inflationary reheating and cosmological perturbations during inflation. Finally, we conjecture that the same rate appears in the entanglement growth of chaotic quantum systems prepared in a semiclassical state.

[10.1007/JHEP03(2018)025] | [arXiv:1709.00427]