Entanglement-area law for general bosonic harmonic lattice systems

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.

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Figure 2

(Color online) Entries of $R$ in one dimension, $D=1$, for a finite-ranged coupling matrix $V$, $C=[1,\dots ,100]$ and $I=[30,\dots ,35]\cup [65,\dots ,70]$, yielding $s\left(I\right)=4$. Bars show $R_{i,j}$; color-encoded surface depicts $\ln \left(R_{i,j}\right)$. All units are arbitrary. Note that the entries decay exponentially away from the boundary of $I$.

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(Color online) The entries ${\left[V^{-1∕2}\right]}_{\mathit{i},\mathit{j}}$ for a finite-ranged coupling matrix $V$ in two dimensions, $D=2$, $\mathit{j}=\mathit{i}+(x_1,x_2)$. Bars show ${\left[V^{-1∕2}\right]}_{\mathit{i},\mathit{j}}$ and the color-encoded surface shows $\ln \left({\left[V^{-1∕2}\right]}_{\mathit{i},\mathit{j}}\right)$. The inset depicts the same for $x_1=x_2$. All units are arbitrary. Note the exponential decay away from the main diagonal $\mathit{i}=\mathit{j}$.

Figure 4

(Color online) Matrix entries of $R$ for the case $D=1$, $C=[1,\dots ,100]$, $I=[1,\dots ,15]$, $V=M^2$, where $M$ is a nearest-neighbor circulant matrix. Note that for this particular form of $V$ the entries $R_{i,j}$ are exactly zero for $j\ne 1,m$. This is in contrast to the general form of $R$ depicted in Fig. 2. The inset shows the entries $R_{i,1}$ as bars and $\ln \left(R_{i,1}\right)$ in blue, cf. Eq. (17).

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