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

• Published in: The journal of high energy physics Vol. 2018; no. 3; pp. 1 - 70
• Main Authors: Bianchi, Eugenio, Hackl, Lucas, Yokomizo, Nelson
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-03-2018; Springer Nature B.V; SpringerOpen
• Subjects: Chaos theory; Classical and Quantum Gravitation; Elementary Particles; Entropy; Field Theories in Lower Dimensions; Field theory; Heating; High energy physics; Inflation; Inflation (cosmology); Lattice Quantum Field Theory; Liapunov exponents; Physics; Physics and Astronomy; Quantum Dissipative Systems; Quantum entanglement; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quantum theory; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Field Theories in Lower Dimensions; Lattice Quantum Field Theory; Quantum Dissipative Systems
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP03(2018)025

A
bstract
The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate
h
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 Λ
A
≤
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 Λ
A
appears in the entanglement growth of chaotic quantum systems prepared in a semiclassical state.