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arXiv:2103.02492 · ACM Symposium on Theory of Computing (STOC)

An area law for 2d frustration-free spin systems

Anurag Anshu, Itai Arad, David Gosset

Published Jun 9, 2022• 40 citations
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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 є by a degree O(√nlog(є−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.

DOI
10.1145/3519935.3519962

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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 є by a degree O(√nlog(є−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.
Let H be a frustration-free Hamiltonian describing a 2D grid of qudits with local interactions, a unique ground state, and local spectral gap lower bounded by a positive constant. For any bipartition defined by a vertical cut of length L running from top to bottom of the grid, we prove that the corresponding entanglement entropy of the ground state of H is upper bounded by Õ(L 5/3). For the special case of a 1D chain, our result provides a new area law which improves upon prior work, in terms of the scaling with qudit dimension and spectral gap. In addition, for any bipartition of the grid into a rectangular region A and its complement, we show that the entanglement entropy is upper bounded as Õ(|∂ A|5/3) where ∂ A is the boundary of A. This represents a subvolume bound on entanglement in frustration-free 2D systems. In contrast with previous work, our bounds depend on the local (rather than global) spectral gap of the Hamiltonian. We prove our results using a known method which bounds the entanglement entropy of the ground state in terms of certain properties of an approximate ground state projector (AGSP). To this end, we construct a new AGSP which is based on a robust polynomial approximation of the AND function and we show that it achieves an improved trade-off between approximation error and entanglement.
We show an area law in the mutual information for the maximally-mixed state Ω in the ground space of general Hamiltonians, which is independent of the underlying ground space degeneracy. Our result assumes the existence of a ‘good’ approximation to the ground state projector (a good AGSP), a crucial ingredient in former area-law proofs. Such approximations have been explicitly derived for 1D gapped local Hamiltonians and 2D frustration-free locally-gapped local Hamiltonians. As a corollary, we show that in 1D gapped local Hamiltonians, for any є>0 and any bi-partition L∪ Lc of the system,
We show an area law in the mutual information for the maximally-mixed state Ω\Omega Ω in the ground space of general Hamiltonians, which is independent of the underlying ground space degeneracy. Our result assumes the existence of a ‘good’ approximation to the ground state projector (a good AGSP), a crucial ingredient in previous area-law proofs. Such approximations have been explicitly derived for 1D gapped local Hamiltonians and 2D frustration-free locally-gapped Hamiltonians. As a corollary, we show that in 1D gapped local Hamiltonians, for any ϵ>0\epsilon >0ϵ>0 and any bipartition L∪LcL\cup L^cL∪Lc of the system, Imax⁡ϵ  ⁣ ⁣(L:Lc)Ω≤O(log⁡(∣L∣log⁡(d))+log⁡(1/ϵ)),\begin{aligned} \textrm{I}^\epsilon _{\max } \, \! \! \left( L : L^c \right) _{\Omega } \le {\textrm{O}}\left( \log (|L|\log (d))+\log (1/\epsilon )\right) , \end{aligned}Imaxϵ​(L:Lc)Ω​≤O(log(∣L∣log(d))+log(1/ϵ)),​ where |L| represents the number of sites in L, d is the dimension of a site and Imax⁡ϵ  ⁣ ⁣(L:Lc)Ω \textrm{I}^\epsilon _{\max } \, \! \! \left( L : L^c \right) _{\Omega }Imaxϵ​(L:Lc)Ω​ represents the ϵ\epsilon ϵ -smoothed maximum mutual information with respect to the L:LcL:L^cL:Lc partition in Ω\Omega Ω . From this bound we then conclude I  ⁣ ⁣(L:Lc)Ω≤O(log⁡(∣L∣log⁡(d))) \textrm{I} \, \! \! \left( L : L^c \right) _\Omega \le {\textrm{O}}\left( \log (|L|\log (d))\right) I(L:Lc)Ω​≤O(log(∣L∣log(d))) – an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that Ω\Omega Ω can be approximated in trace norm up to ϵ\epsilon ϵ with a state of Schmidt rank of at most poly(∣L∣log⁡(d)/ϵ)\textrm{poly}(|L|\log (d)/\epsilon )poly(∣L∣log(d)/ϵ) , leading to a good MPO approximation for Ω\Omega Ω with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians.
We show an area law in the mutual information for the maximally-mixed state ΩΩΩ in the ground space of general Hamiltonians, which is independent of the underlying ground space degeneracy. Our result assumes the existence of a `good' approximation to the ground state projector (a good AGSP), a crucial ingredient in previous area-law proofs. Such approximations have been explicitly derived for 1D gapped local Hamiltonians and 2D frustration-free locally-gapped Hamiltonians. As a corollary, we show that in 1D gapped local Hamiltonians, for any ε>0\varepsilon>0ε>0 and any bi-partition L∪LcL\cup L^cL∪Lc of the system, \begin{align*} \mathrm I_{\max}^\varepsilon (L:L^c)_Ω \le \mathrm O \big( \log (|L|\log(d))+\log(1/\varepsilon)\big), \end{align*} where ∣L∣|L|∣L∣ represents the number of sites in LLL, ddd is the dimension of a site and $ \mathrm I_{\max}^\varepsilon (L:L^c)_Ω $ represents the ε\varepsilonε-\emph{smoothed maximum mutual information} with respect to the L:LcL:L^cL:Lc partition in ΩΩΩ. From this bound we then conclude I(L:Lc)Ω≤O(log⁡(∣L∣log⁡(d)))\mathrm I (L:L^c)_Ω \le \mathrm O\big(\log(|L|\log(d))\big)I(L:Lc)Ω​≤O(log(∣L∣log(d))) -- an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that ΩΩΩ can be approximated in trace norm up to ε\varepsilonε with a state of Schmidt rank of at most poly(∣L∣/ε)\mathrm{poly}(|L|/\varepsilon)poly(∣L∣/ε), leading to a good MPO approximation for ΩΩΩ with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians.
We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Z{sup d}. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset {chi} {contained_in} V the ground state projector can be approximated by the product of two projections, one supported on {chi} and one supported on {chi}{sup c}, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. Such an approximation was useful in proving an area law in one dimension, and this result corresponds to a multi-dimensional analogue.
We prove an entanglement area law for a class of 1D quantum systems involving infinite-dimensional local Hilbert spaces. This class of quantum systems include bosonic models such as the Hubbard-Holstein model, and both U(1) and SU(2) lattice gauge theories in one spatial dimension. Our proof relies on new results concerning the robustness of the ground state and spectral gap to the truncation of Hilbert space, applied within the approximate ground state projector (AGSP) framework from previous work. In establishing this area law, we develop a system-size independent bound on the expectation value of local observables for Hamiltonians without translation symmetry, which may be of separate interest. Our result provides theoretical justification for using tensor network methods to study the ground state properties of quantum systems with infinite local degrees of freedom.