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.
Highlights
Entanglement is a cornerstone of quantum theory and its dynamics has been extensively studied in a wide range of different systems [1,2,3,4,5]
The present paper combines these recent findings of [21] with the insights about the entanglement entropy growth for Gaussian states of [18]. This allows us to treat the most general case of an arbitrary time-dependent quadratic Hamiltonian H (t) and an arbitrary pure initial state ρ of a bipartite bosonic quantum system AB, for which we prove that the entanglement entropy grows as
The main ingredient of the lower bound for the time scaling of the entanglement entropy is the generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems proved in [21]
Summary
Entanglement is a cornerstone of quantum theory and its dynamics has been extensively studied in a wide range of different systems [1,2,3,4,5]. Let us emphasize that while our rigorous results describe the long-time asymptotics t → ∞ of quadratic Hamiltonians, we will discuss their physical significance and applications in the context of interacting and periodically driven systems, where this asymptotics describes an intermediate phase before the entanglement entropy eventually saturates This manuscript is structured as follows: In section 2, we first review the results of [18] and [21] in order to prove the required propositions for our main results, i.e., the linear growth of the entanglement entropy for arbitrary pure initial states, and its extension to squashed entanglement of mixed initial states.
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