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arXiv:quant-ph/0410246 · Physical Review A

Entanglement across a transition to quantum chaos

Carlos Mejía-Monasterio, Giuliano Benenti, Gabriel G. Carlo, Giulio Casati

Published Jun 1, 2005• 75 citations
arXivHTMLPDF

Abstract

We study the relation between entanglement and quantum chaos in one- and two-dimensional spin-$1∕2$ lattice models, which exhibit mixing of the noninteracting eigenfunctions and transition from integrability to quantum chaos. Contrary to what occurs in a quantum phase transition, the onset of quantum chaos is not a property of the ground state but takes place for any typical many-spin quantum state. We study bipartite and pairwise entanglement measures---namely, the reduced von Neumann entropy and the concurrence---and discuss quantum entanglement sharing. Our results suggest that the behavior of the entanglement is related to the mixing of the eigenfunctions rather than to the transition to chaos.

DOI
10.1103/physreva.71.062324

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We investigate disordered oneand two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators such as the number of principal components is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spinand fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.
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It is widely recognized that entanglement generation and dynamical chaos are intimately related in semiclassical models via the process of decoherence. In this paper, we propose a unifying framework which directly connects the bipartite and multipartite entanglement growth to the quantifiers of classical and quantum chaos. In the semiclassical regime, the dynamics of the von Neumann entanglement entropy, the spin squeezing, the quantum Fisher information, and the out-of-time-order square commutator are governed by the divergence of nearby phase-space trajectories via the local Lyapunov spectrum, as suggested by previous conjectures in the literature. General analytical predictions are confirmed by detailed numerical calculations for two paradigmatic models, relevant in atomic and optical experiments, which exhibit a regular-to-chaotic transition: the quantum kicked top and the Dicke model.
We show that in systems with highly degenerate energy spectra, such as the two-dimensional transverse-field Ising model (2DTFIM) in the strong-field limit, quantum chaos can emerge in finite systems for arbitrary small perturbations. In this regime, the presence of extensive quasiconserved quantities can prevent finite systems from becoming ergodic. We study the ensuing crossover to ergodicity in a family of models that includes the 2DTFIM, in which the onset of ergodic behavior exhibits universality and occurs for perturbation strengths that decrease polynomially with increasing system size. We discuss the behaviors of quantum chaos indicators, such as level spacing statistics and bipartite entanglement, and of the fidelity susceptibilities and spectral functions across the crossover.
We study nonequilibrium processes in an isolated quantum system-the Dicke model-focusing on the role played by the transition from integrability to chaos and the presence of excited-state quantum phase transitions. We show that both diagonal and entanglement entropies are abruptly increased by the onset of chaos. Also, this increase ends in both cases just after the system crosses the critical energy of the excited-state quantum phase transition. The link between entropy production, the development of chaos, and the excited-state quantum phase transition is more clear for the entanglement entropy.