An area 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 $\epsilon$ by a degree $O(\sqrt{n\log ({\epsilon }^{-1})})$ 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.

STOC 2022; version 3: fixed the choice of beta on Page 16. Results unchanged. 25 pages, 3 figures

References in corpus (6)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Black holes as mirrors: quantum information in random subsystems
• Entropy and Entanglement in Quantum Ground States
• Local tests of global entanglement and a counterexample to the generalized area law
• Sufficient Condition for Entanglement Area Laws in Thermodynamically Gapped Spin Systems
• A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions

Cited by in corpus (24)

• Speed limits and locality in many-body quantum dynamics
• NLTS Hamiltonians from good quantum codes
• Bounds in Nonequilibrium Quantum Dynamics
• Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle
• Fault-tolerant quantum algorithms for quantum molecular systems: A survey
• Matrix Product Operator Algebras II: Phases of Matter for 1D Mixed States
• The frustration-free fully packed loop model
• Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations
• Quantitatively improved finite-size criteria for spectral gaps
• Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models
• Random translation-invariant Hamiltonians and their spectral gaps
• The dynamical $\alpha$-Rényi entropies of local Hamiltonians grow at most linearly in time
• Introduction to quantum entanglement in many-body systems
• Thermal Area Law for Lattice Bosons
• The resource theory of tensor networks
• Area law for steady states of detailed-balance local Lindbladians
• Bicolor loop models and their long range entanglement
• A construction of Combinatorial NLTS
• Quantum lozenge tiling and entanglement phase transition
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
• Area laws and tensor networks for maximally mixed ground states
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• Sequential Generation of Two-dimensional Super-area-law States with Local Parent Hamiltonian
• Energy Spectra of Compressed Quantum States