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arXiv:quant-ph/0609044 · Physical Review A

Entanglement of collectively interacting harmonic chains: An effective two-dimensional system

Razmik G. Unanyan, Michael Fleischhauer, Dagmar Bruß

Published Apr 27, 2007• 7 citations
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Abstract

We study the ground-state entanglement of one-dimensional harmonic chains that are coupled to each other by a collective interaction as realized, e.g., in an anisotropic ion crystal. Due to the collective type of coupling, where each chain interacts with every other one in the same way, the total system shows critical behavior in the direction orthogonal to the chains, while the isolated harmonic chains can be gapped and noncritical. We derive lower and most importantly upper bounds for the entanglement, quantified by the von Neumann entropy, between a compact block of oscillators and its environment. For sufficiently large size of the subsystems, the bounds coincide and show that the area law for entanglement is violated by a logarithmic correction.

DOI
10.1103/physreva.75.040302

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We study the ground-state entanglement of one-dimensional harmonic chains that are coupled to each other by a collective interaction as realized, e.g., in an anisotropic ion crystal. Due to the collective type of coupling, where each chain interacts with every other one in the same way, the total system shows critical behavior in the direction orthogonal to the chains, while the isolated harmonic chains can be gapped and noncritical. We derive lower and most importantly upper bounds for the entanglement, quantified by the von Neumann entropy, between a compact block of oscillators and its environment. For sufficiently large size of the subsystems, the bounds coincide and show that the area law for entanglement is violated by a logarithmic correction.
We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic-lattice system in the ground or a thermal state scale at most as the boundary area of the region. This area law is rigorously proven to hold true in noncritical harmonic-lattice systems of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape, and states of nonzero temperature. For nearest-neighbor interactions---corresponding to the Klein-Gordon case---upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of Phys. Rev. Lett. 94, 060503 (2005). These higher-dimensional analogs of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in noncritical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block-banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two-point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement.
We study the long-time evolution of the bipartite entanglement in translationally invariant gapped harmonic lattice systems with finite-range interactions. A lower bound for the von Neumann entropy is derived in terms of the purity of the reduced density matrix. It is shown that starting from an initially Gaussian state the entanglement entropy increases at least linearly in time. This implies that the dynamics of gapped (non-critical) harmonic lattice systems cannot be efficiently simulated by algorithms based on matrix-product decompositions of the quantum state.
We study the long-time evolution of the bipartite entanglement in translationally invariant one-dimensional harmonic lattice systems. We show that for Gaussian states, in quadratic interactions with periodic boundary conditions, there exists a lower bound for the von Neumann entropy which increases linearly in time. This implies that the dynamics of harmonic lattice systems can in general not efficiently be simulated by algorithms based on matrix-product decompositions of the quantum state, and interactions are needed to suppress the entanglement growth with time.
We study different aspects of quantum von Neumann and R\'enyi entanglement entropy of one-dimensional long-range harmonic oscillators that can be described by well-defined nonlocal field theories. We show that the entanglement entropy of one interval with respect to the rest changes logarithmically with the number of oscillators inside the subsystem. This is true also in the presence of different boundary conditions. We show that the coefficients of the logarithms coming from different boundary conditions can be reduced to just two different universal coefficients. We also study the effect of the mass and temperature on the entanglement entropy of the system in different situations. The universality of our results is also confirmed by changing different parameters in the coupled harmonic oscillators. We also show that more general interactions coming from general singular Toeplitz matrices can be decomposed to our long-range harmonic oscillators. Despite the long-range nature of the couplings, we show that the area law is valid in two dimensions and the universal logarithmic terms appear if we consider subregions with sharp corners. Finally, we study analytically different aspects of the mutual information such as its logarithmic dependence to the subsystem, effect of mass, and influence of the boundary. We also generalize our results in this case to general singular Toeplitz matrices and higher dimensions.
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Bipartite entanglement in the ground state of a chain of NNN quantum spins can be quantified either by computing pairwise concurrence or by dividing the chain into two complementary subsystems. In the latter case the smaller subsystem is usually a single spin or a block of adjacent spins and the entanglement differentiates between critical and noncritical regimes. Here we extend this approach by considering a more general setting: our smaller subsystem SA{S}_{A}SA​ consists of a comb of LLL spins, spaced ppp sites apart. Our results are thus not restricted to a simple area law, but contain nonlocal information, parametrized by the spacing ppp. For the XXXXXX model we calculate the von Neumann entropy analytically when $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and investigate its dependence on LLL and ppp. We find that an external magnetic field induces an unexpected length scale for entanglement in this case.