56 research outputs found
Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate
The rate of entropy production in a classical dynamical system is
characterized by the Kolmogorov-Sinai entropy rate 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
of a Gaussian state grows linearly for large times in unstable systems,
with a rate determined by the Lyapunov exponents and
the choice of the subsystem . 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 appears in the entanglement growth of
chaotic quantum systems prepared in a semiclassical state.Comment: 50+17 Pages, 11 figure
Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
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
Circuit complexity for free fermions
We study circuit complexity for free fermionic field theories and Gaussian
states. Our definition of circuit complexity is based on the notion of geodesic
distance on the Lie group of special orthogonal transformations equipped with a
right-invariant metric. After analyzing the differences and similarities to
bosonic circuit complexity, we develop a comprehensive mathematical framework
to compute circuit complexity between arbitrary fermionic Gaussian states. We
apply this framework to the free Dirac field in four dimensions where we
compute the circuit complexity of the Dirac ground state with respect to
several classes of spatially unentangled reference states. Moreover, we show
that our methods can also be applied to compute the complexity of excited
states. Finally, we discuss the relation of our results to alternative
approaches based on the Fubini-Study metric, the relevance to holography and
possible extensions.Comment: 84 pages, 10 figures, 1 tabl
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