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arXiv:quant-ph/0506022 · Physical Review A

Entanglement of bosonic modes in symmetric graphs

M. Asoudeh, Vahid Karimipour

Published Sep 30, 2005• 3 citations
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Abstract

The ground and thermal states of a quadratic Hamiltonian representing the interaction of bosonic modes or particles are always Gaussian states. We investigate the entanglement properties of these states for the case where the interactions are represented by harmonic forces acting along the edges of symmetric graphs---i.e., one-, two-, and three-dimensional rectangular lattices, mean-field clusters, and platonic solids. We determine the entanglement of formation (EOF) as a function of the interaction strength, calculate the maximum EOF in each case, and compare these values with the bounds found previously for quadratic Hamiltonians.

DOI
10.1103/physreva.72.032339

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We consider a collection of bosonic modes corresponding to the vertices of a graph Γ. Quantum tunneling can occur only along the edges of Γ and a local self-interaction term is present. Quantum entanglement of one vertex with respect to the rest of the graph (mode entanglement) is analyzed in the ground state of the system as a function of the tunneling amplitude τ. The topology of Γ plays a major role in determining the tunneling amplitude τ max that leads to the maximum value of the mode entanglement. Whereas in most of the cases one finds the intuitively expected result τ max = ∞ , we show that there exists a family of graphs for which the optimal value of τ is pushed down to a finite value. We also show that, for complete graphs, our bi-partite entanglement provides useful insights in the analysis of the crossover between insulating and superfluid ground states.
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In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous variable entanglement for a system consisting of two non-interacting bosonic modes embedded in a thermal environment. The calculated measure of entanglement is entanglement of formation. We describe the evolution of entanglement in terms of the covariance matrix for symmetric Gaussian input states. In the case of an entangled initial squeezed thermal state, entanglement suppression (entanglement sudden death) takes place, for all non-zero temperatures of the thermal bath. After that, the system remains for all times in a separable state. For a zero temperature of the thermal bath, the system remains entangled for all finite times, but in the limit of asymptotic large times the state becomes separable.
This article reviews and extends recent results concerning entanglement and frustration in multipartite systems which have some symmetry with respect to the ordering of the particles. Starting point of the discussion are Bell inequalities: their relation to frustration in classical systems and their satisfaction for quantum states which have a symmetric extension. We then discussed how more general global symmetries of multipartite systems constrain the entanglement between two neighboring particles. We prove that maximal entanglement (measured in terms of the entanglement of formation) is always attained for the ground state of a certain nearest neighbor interaction Hamiltonian having the considered symmetry with the achievable amount of entanglement being a function of the ground state energy. Systems of Gaussian states, i.e. quantum harmonic oscillators, are investigated in more detail and the results are compared to what is known about ordered qubit systems.