Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

TL;DR

This paper proves that for any pure initial state in a bipartite bosonic system, the entanglement entropy grows linearly over time under unstable quadratic Hamiltonian dynamics, with a rate determined by classical Lyapunov exponents.

Contribution

It generalizes previous Gaussian-specific results to arbitrary initial states using a new entropy subadditivity technique, applicable to mixed states and broader physical systems.

Findings

Entanglement entropy grows linearly with time for unstable quadratic Hamiltonians.

Growth rate equals the sum of classical Lyapunov exponents, independent of initial state.

Results extend to mixed states and various physical models, including quantum field theories.

Abstract

We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the initial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement entropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models.