  The rate of entropy production in a classical dynamical system is
characterized by the Kolmogorov-Sinai entropy rate $h_{\mathrm{KS}}$ 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
$S_A$ of a Gaussian state grows linearly for large times in unstable systems,
with a rate $\Lambda_A \leq h_{KS}$ 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 $\Lambda_A$ appears in the entanglement growth of
chaotic quantum systems prepared in a semiclassical state.
