An entanglement-area law for general bosonic harmonic lattice systems

Abstract

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 non-critical harmonic lattice system 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 analogues 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 non-critical 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.

Figures (5)

• Figure 1: A two-dimensional lattice system $C=[1,...,n]^{\times 2}$ with a distinguished (shaded) region $I$, consisting of $v(I)$ degrees of freedom . Oscillators representing the exterior $O=C\setminus I$ are shown as , whereas pairs belonging to the surface $s(I)$ of the region are marked by lines ().
• Figure 2: Entries of $R$ in one dimension, $D=1$, for a finite-ranged coupling matrix $V$, $C=[1,...,100]$ and $I=[30,...,35]\cup[65,...,70]$, yielding $s(I)=4$. Bars show $R_{i,j}$, color-encoded surface depicts $\log(R_{i,j})$. All units are arbitrary. Note that the entries decay exponentially away from the boundary of $I$.
• Figure 3: The entries $[V^{-1/2}]_{\boldsymbol{i},\boldsymbol{j}}$ for a finite-ranged coupling matrix $V$ in two dimensions, $D=2$, $\boldsymbol{j}=\boldsymbol{i}+(x_1,x_2)$. Bars show $[V^{-1/2}]_{\boldsymbol{i},\boldsymbol{j}}$ and the color-encoded surface shows $\log([V^{-1/2}]_{\boldsymbol{i},\boldsymbol{j}})$. The inset depicts the same for $x_1=x_2$. All units are arbitrary. Note the exponential decay away from the main diagonal $\boldsymbol{i}=\boldsymbol{j}$.
• Figure 4: Matrix entries of $R$ for the case $D=1$, $C=[1,...,100]$, $I=[1,...,15]$, $V=M^2$, 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. \ref{['R']}. The inset shows the entries $R_{i,1}$ as bars and $\log(R_{i,1})$ in blue (cf. Eq. (\ref{['Rdecay']})).
• Figure 5: Visualization of the enumeration of $N_{l}$ as in Appendix \ref{['abzaehlerei']}. As before, oscillators belonging to the distinguished (shaded) region $I$ are marked , the outside ones are shown as . $\partial O$ is shown as the orange shaded area. $M_{4}(\boldsymbol{i})=\{\boldsymbol{j}\in C|d(\boldsymbol{i},\boldsymbol{j})\leq 4\}$ for a certain oscillator is shaded green, its surface oscillators $m_{4}(\boldsymbol{i})=\{\boldsymbol{j}\in C|d(\boldsymbol{i},\boldsymbol{j}) = 4\}$ are depicted by .