An area law for one-dimensional quantum systems

By Erwin Schrödinger International, M. B. Hastings. 2007

1,091 citations recorded by the source corpus.

We prove an area law for the entanglement entropy in gapped one-dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on matrix product states which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on Rényi entropy for sufficiently large α<1 and implies the ability to approximate the ground state by a matrix product state.

View Paper: https://www.semanticscholar.org/paper/b53cb499a37b7cd213db4f8d593d14c3761f4883

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Citations in the corpus

The density matrix renormalization group for ab initio quantum chemistry (2014, background): “This is a theorem for onedimensional systems [33] and a conjecture in higher dimensions [52], supported by numerical examples and theoretical arguments [53].”
Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems (2011, unknown): “The quasi-adiabatic continuation technique was subsequently elaborated upon and used in new applications by Hastings and collaborators [9, 20, 23, 8, 7] as well as other authors [40, 44].”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, methodology): “In the case of gapped quantum spin chains in one dimension, a remarkable theorem has been formulated by Hastings (2007) proving the area law which was later strengthened by Arad, Kitaev, Landau, and Vazirani (2013): given a local Hamiltonian of a quantum spin chain of N ddimensional spins whose gap…”
Adiabatic Quantum Computing (2016, background): “…a finite length, which leads to area laws for the entanglement entropy (Eisert et al., 2010).38 E.g., the area law for gapped 1D systems, proved in (Hastings, 2007), states that for the ground state, the entanglement of any interval is upper bounded by a constant independent of the size of the…”
Computations in quantum tensor networks (2012, background): “In one-dimensional systems, a rigorous area law has recently been proven for all systems with a gap between the smallest and the second smallest eigenvalue [17].”
Quantum spin liquid states (2016, background): “In practice, the low-energy states of gapped local Hamiltonians in one dimension can be efficiently represented by MPSs with a finite value of M (Hastings, 2007; Verstraete and Cirac, 2006).”
Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy) (2018, background): “uctuations within the correlation length ˘and thus only degrees of freedom near the cut are entangled, as schematically indicated in Fig. 1(b). A rigorous proof of the area law in 1D is given in Ref. [10]. In this respect, ground states are very special states and can be found within a very small co”
Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases (2021, background): “It is known that up to exponentially small errors in local quantities, ground states of gapped Hamiltonians are well approximated by such an MPS [122, 123].”