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

  • Eugenio Bianchi
  • Lucas HacklEmail author
  • Nelson Yokomizo
Open Access
Regular Article - Theoretical Physics


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 ΛAhKS 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.


Field Theories in Lower Dimensions Lattice Quantum Field Theory Quantum Dissipative Systems 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute for Gravitation and the Cosmos & Department of Physics, The Pennsylvania State University, Davey LaboratoryUniversity ParkU.S.A.
  2. 2.Departamento de Física — ICEx, Universidade Federal de Minas GeraisBelo HorizonteBrazil

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