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arXiv:1111.2970 · Physical Review B

Improved one-dimensional area law for frustration-free systems

Itai Arad, Zeph A. Landau, Umesh V. Vazirani

Published May 29, 2012• 77 citations
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Abstract

We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings\^a 1D area law and which is tight to within a polynomial factor. For particles of dimension $d$, spectral gap $\ensuremath{\epsilon}>0$, and interaction strength at most $J$, our entropy bound is ${S}_{1D}\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}X$, where $X\stackrel{\mathrm{def}}{=}(J\mathrm{log}d)/\ensuremath{\epsilon}$. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. In higher dimensions, when the bipartitioning area is $|\ensuremath{\partial}L|$, we use additional local structure in the proof and show that $S\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{|\ensuremath{\partial}L|}^{2}{\mathrm{log}}^{6}|\ensuremath{\partial}L|\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}X$. This is at the cusp of being nontrivial in the 2D case, in the sense that any further improvement would yield a subvolume law.

DOI
10.1103/physrevb.85.195145

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We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings^a 1D area law and which is tight to within a polynomial factor. For particles of dimension ddd, spectral gap \ensuremathϵ>0\ensuremath{\epsilon}>0\ensuremathϵ>0, and interaction strength at most JJJ, our entropy bound is S1D\ensuremath≤O(1)\ifmmode⋅\else\textperiodcentered\fiX3log8X{S}_{1D}\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}XS1D​\ensuremath≤O(1)\ifmmode⋅\else\textperiodcentered\fiX3log8X, where X=def(Jlogd)/\ensuremathϵX\stackrel{\mathrm{def}}{=}(J\mathrm{log}d)/\ensuremath{\epsilon}X=def(Jlogd)/\ensuremathϵ. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. In higher dimensions, when the bipartitioning area is ∣\ensuremath∂L∣|\ensuremath{\partial}L|∣\ensuremath∂L∣, we use additional local structure in the proof and show that S\ensuremath≤O(1)\ifmmode⋅\else\textperiodcentered\fi∣\ensuremath∂L∣2log6∣\ensuremath∂L∣\ifmmode⋅\else\textperiodcentered\fiX3log8XS\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{|\ensuremath{\partial}L|}^{2}{\mathrm{log}}^{6}|\ensuremath{\partial}L|\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}XS\ensuremath≤O(1)\ifmmode⋅\else\textperiodcentered\fi∣\ensuremath∂L∣2log6∣\ensuremath∂L∣\ifmmode⋅\else\textperiodcentered\fiX3log8X. This is at the cusp of being nontrivial in the 2D case, in the sense that any further improvement would yield a subvolume law.
We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result \cite{ref:Has07}. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \eps>0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by $O{\frac{\log^3 d}{\eps}}$. Our approach uses the framework Arad et al to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the $1/\eps$ dependence of the bound in Arad et al. To achieve that, we establish a new, "random-walk like", bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: \ER{H^\ell} \le (\ell d)^{O(\sqrt{\ell})}. Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension $B=e^{O(\log^{3/4}n/\eps^{1/4})}. Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1/\poly(n) approximation of the ground energy with a subexponential running time T\le \exp(e^{O(\log^{3/4}n/\eps^{1/4})}).
The classical description of quantum states is in general exponential in the number of qubits. Can we get polynomial descriptions for more restricted sets of states such as ground states of interesting subclasses of local Hamiltonians? This is the basic problem in the study of the complexity of ground states, and requires an understanding of multi-particle entanglement and quantum correlations in such states. Area laws provide a fundamental ingredient in the study of the complexity of ground states, since they offer a way to bound in a quantitative way the entanglement in such states. Although they have long been conjectured for many body systems in arbitrary dimensions, a general rigorous was only recently proved in Hastings' seminal paper [8] for ID systems. In this paper, we give a combinatorial proof of the ID area law for the special case of frustration free systems, improving by an exponential factor the scaling in terms of the inverse spectral gap and the dimensionality of the particles. The scaling in terms of the dimension of the particles is a potentially important issue in the context of resolving the 2D case and higher dimensions, which is one of the most important open questions in Hamiltonian complexity. Our proof is based on a reformulation of the detectability lemma, introduced by us in the context of quantum gap amplification [1]. We give an alternative proof of the detectability lemma, which is not only simpler and more intuitive than the original proof, but also removes a key restriction in the original statement, making it more suitable for this new context. We also give a one page proof of Hastings' proof that the correlations in the ground states of gapped Hamiltonians decay exponentially with the distance, demonstrating the simplicity of the combinatorial approach for those problems.
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.
Knabe's theorem lower bounds the spectral gap of a one-dimensional frustration-free local Hamiltonian in terms of the local spectral gaps of finite regions. It also provides a local spectral gap threshold for Hamiltonians that are gapless in the thermodynamic limit, showing that the local spectral gap must scale inverse linearly with the length of the region for such systems. Recent works have further improved upon this threshold, tightening it in the one-dimensional case and extending it to higher dimensions. Here, we show a local spectral gap threshold for frustration-free Hamiltonians on a finite-dimensional lattice that is optimal up to a constant factor that depends on the dimension of the lattice. Our proof is based on the detectability lemma framework and uses the notion of a coarse-grained Hamiltonian (introduced in [Anshu et al., Phys. Rev. B 93, 205142]) as a link connecting the (global) spectral gap and the local spectral gap.
The Rényi area law conjecture is false below α=1α=1α=1 in high dimensions
Area laws express the expectation that entanglement in gapped quantum ground states scales with a boundary rather than a volume. In one dimension this holds for every Rényi entropy, and the Rényi area-law conjecture asserts the same in higher dimensions for each fixed 0<α<10<α<10<α<1. Here we disprove this conjecture in every spatial dimension two and higher. We construct finite-range Hamiltonians with fixed local dimension, bounded interaction strength, unique ground states and a uniform spectral gap, whose ground states exhibit volume-law Rényi entanglement while obeying a von Neumann area law and remaining connected to product states through uniformly gapped local paths. The mechanism is error-space complexity: locally identifiable components of arbitrarily small weight can encode states with high entanglement complexity. We also establish the worst-case rank-error trade-off for approximate ground-state projectors, a key tool in one-dimensional area-law proofs. At fixed gap, the optimal Schmidt rank at projection error εεε across a planar cut of cross-section AAA scales as exp⁡[Θ~((Alog⁡ε−1)2/3)]\exp[\tildeΘ((A\logε^{-1})^{2/3})]exp[Θ~((Alogε−1)2/3)] when log⁡ε−1≳A\logε^{-1}\gtrsim Alogε−1≳A. Thus α=1α=1α=1 is the sharp threshold for a universal Rényi area law within this adiabatically connected class.
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.