Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate

Abstract

The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate hKS 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 SA of a Gaussian state grows linearly for large times in unstable systems, with a rate ΛA ≤ hKS determined by the Lyapunov exponents and the choice of the subsystem A. 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 ΛA appears in the entanglement growth of chaotic quantum systems prepared in a semiclassical state.