Entanglement subvolume law for 2d frustration-free spin systems

Abstract

We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error ǫ by a degree O( n log(ǫ −1 )) multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special case of commuting Hamiltonian terms. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation in the vicinity of the boundary of the bipartition of interest. This AGSP has sufficiently low entanglement and error to establish the area law using a known technique.

Area law of noncritical ground states in 1D long-range interacting systems

“…where we use k jψ p i À 0 j i k ≤ γ p . By applying inequalities (16) and (17) to Claim 8, we can achieve k 0 j i À jψ D i k 2 ≤ 2nδ;…”

“…However, providing detailed proof of the area law is still an extremely challenging problem. So far, the proof of this law is limited to gapped 1D systems [4][5][6][7] , 1D quantum states with finite correlation lengths 8,9 , gapped harmonic lattice systems 10,11 , tree-graph systems 12 , and high-dimensional systems with specific assumptions [13][14][15][16][17] (see ref. 3 for a comprehensive review).…”

Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states

“…A similar notion also applies to finite temperature systems. Although a rigorous proof of the area law at zero temperatures appears to be a notoriously challenging problem [20][21][22][23][24][25][26][27], an analogous area law at finite temperatures has been proved by Wolf et al [28] in a simple and elegant manner. The authors proved the following inequality:…”

“…Now, we discuss the key principles that allow us to improve the original thermal area law (See Appendix C for the details). Our analysis utilizes various recent techniques employed in the proofs of the area law for ground states [23,26,27]. Inspired by these studies, we construct an approximation of the quantum Gibbs state using an appropriate polynomial of low degree [89] and then perform a Schmidt rank analysis adapted from [23].…”

“…We apply an analysis similar to that in Ref. [26], which modified the proof in [23] for the Schmidt rank estimation. First, we decompose S into l 0 blocks {B s } l0+1 s=0 with |B s | = k (s = 1, 2, .…”