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arXiv:0705.3026 · Quantum Information and Computation

Entanglement and separability of quantum harmonic oscillator systems at finite temperature

Janet Anders, Andreas Winter

Published Mar 1, 2008• 46 citations
arXivHTMLPDF

Abstract

In the present paper we study the entanglement properties of thermal (a.k.a.~\emph{Gibbs}) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system and quantify the degree of entanglement below the critical temperature. Finally, we discuss the example of a ring topology in detail and compare our results with previous work in an entanglement-phase diagram.

DOI
10.26421/qic8.3-4-2

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In the present paper we study the entanglement properties of thermal (a.k.a.~\emph{Gibbs}) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system and quantify the degree of entanglement below the critical temperature. Finally, we discuss the example of a ring topology in detail and compare our results with previous work in an entanglement-phase diagram.
Abstract We present an exact analytical solution of the Hu–Paz–Zhang master equation in a precise Markovian limit for a system of two harmonically coupled harmonic oscillators interacting with a common thermal bath of harmonic oscillators. The thermal bath is initially considered to be at arbitrary temperatures and characterized by an Ohmic Lorentz–Drude spectral density. In the examined system, couplings between the two harmonic oscillators and the environment ensure a complete decoupling of the center-of-mass and relative degrees of freedom, resulting in undamped dynamics in the relative coordinate. The exact time evolution is used to analyze the system’s entanglement dynamics, quantified through logarithmic negativity and quantum mutual information, while ensuring the positivity of the density operator to confirm the physical validity of the results. We demonstrate that, under certain parameter regimes and initial conditions, the asymptotic dynamics can give rise to periodic entanglement-disentanglement behavior. Furthermore, numerical simulations reveal that for negative values of the direct coupling between the oscillators, which are sufficiently close to a critical lower bound beyond which the system becomes unstable, the system can maintain entanglement across a broad temperature range and for arbitrarily long durations.
A meaningful description of the low-energy dynamics of multipartite entanglement is provided\nfor harmonic oscillator systems in general dissipative scenarios. Without doing any\ncentral approximation, this mainly relies on a set of reasonable physical statements on the\nenvironment and system-environment interaction consistent with a linear analysis of the\nopen-system dynamics.\nWe rst investigate the inseparability properties of arbitrary entangled states, which\ngenerally entails an optimization procedure of certain functional de ned in the in nitedimensional\nHilbert space of continuous-variable systems. Using a Gaussian-like assumption\nanalogous to that used in the derivation of Gaussian entanglement of formation, we\nderive a computational-e cient form of such functional which considerable simpli es the\noptimization task. This consists on a hierarchy of separability criteria that permit in a\nuni ed way to characterize k-partite entanglement of broad classes of Gaussian and non-\nGaussian states. The strength of the criteria is proven by showing that they satisfactorily\nreproduce previous results from the theory of entanglement, like PPT criterion applied to\narbitrary two-mode and pure three-mode Gaussian states.\nThe separability criteria allow to monitor the transient evolution of multipartite entanglement\nunder the environmental in\nuence, this permitted to show that non-Gaussian\nentanglement may be as robust against harmful dissipative e ects and thermal noise as\nGaussian one. We devote special attention to the stationary properties of tripartite entanglement\nwhen the system oscillators are in contact with either a common environment\nor independent environments at initial di erent temperatures. For the former dissipative\nscheme, it is shown that the environment mediates an e ective many-party interaction with\na spatial long-range feature between system oscillators, which is able to generate tripartite\nentanglement whereas two-mode entanglement is degraded. Regarding the second scheme,\nit is illustrated that thermal non-equilibrium conditions result in a rich variety of quantum\ncorrelations, however a temperature gradient eventually destroys the entanglement shared\nby the system oscillators as a consequence of the growth of thermal noise. Finally, the\nstationary entanglement and energy current across a nite harmonic chain are studied in\ndetail by doing an extensive numerical analysis of both phenomena for a broad range of the\nparameters (i.e., temperatures, oscillator frequencies, and oscillator coupling strengths).\nOur numerical ndings are discussed in terms of the derived energy current expressions,\nthese show an explicit dependence on the two-time correlation functions (between oscillator\noperators) which carry quantum correlations.
We investigate the entanglement properties of thermal states of the harmonic lattice in one, two, and three dimensions. We establish the value of the critical temperature for entanglement between neighboring sites and give physical reasons. Further sites are shown to be entangled only due to boundary effects. Other forms of entanglement are addressed in the second part of the paper by using the energy as witness of entanglement. We close with a comprehensive diagram showing the different phases of entanglement versus complete separability and propose techniques to swap and tune entanglement experimentally.
Understanding how microscopic few-body interactions give rise to thermal behavior in isolated quantum many-body systems remains a central challenge in nonequilibrium statistical mechanics. While individual energy eigenstates are expected to reproduce thermal equilibrium values, analytic access to highly entangled thermal eigenstates of nonintegrable Hamiltonians remains scarce. In this Letter, we construct exact infinite-temperature eigenstates of generically nonintegrable two-body Hamiltonians using stabilizer states. These states can fully reproduce thermal expectation values for all spatially local observables, extending previously known Bell-pair-based constructions to a broader class. At the same time, we prove a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy microscopic thermal equilibrium for all four-body observables. This bound is tight, as we explicitly construct a translationally invariant Hamiltonian whose stabilizer eigenstate is thermal for all two-body and three-body observables as well as all spatially local observables. Our results suggest that reproducing higher-order thermal correlations requires nonstabilizer degrees of freedom, providing analytic insight into the interplay between interaction locality, microscopic thermal equilibrium, and quantum computational complexity.
In this article we extend the results presented in Ref. [Phys. Rev. A 76, 032101 (2007)] to treat quantitatively the effects of reservoirs at finite temperature in a bosonic dissipative network: a chain of coupled harmonic oscillators whichever its topology, i.e., whichever the way the oscillators are coupled together, the strength of their couplings and their natural frequencies. Starting with the case where distinct reservoirs are considered, each one coupled to a corresponding oscillator, we also analyze the case where a common reservoir is assigned to the whole network. Master equations are derived for both situations and both regimes of weak and strong coupling strengths between the network oscillators. Solutions of these master equations are presented through the normal ordered characteristic function. We also present a technique to estimate the decoherence time of network states by computing separately the effects of diffusion and the attenuation of the interference terms of the Wigner function. A detailed analysis of the diffusion mechanism is also presented through the evolution of the Wigner function. The interesting collective diffusion effects are discussed and applied to the analysis of decoherence of a class of network states. Finally, the entropy and the entanglement of a pure bipartite system are discussed.