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An Area Law for One Dimensional Quantum Systems

M. B. Hastings

TL;DR

The paper proves a rigorous area law for entanglement entropy in one-dimensional gapped quantum systems, deriving a bound S_max that scales exponentially with the correlation length. It introduces a boundary-operator framework (O_B,O_L,O_R) grounded in Lieb-Robinson bounds and a spectral gap, plus a bootstrap argument on block entropies that yields a finite entropy bound and connects to matrix product state representations. It also discusses why data-hiding and quantum expander constructions can complicate naive expectations, and presents a conjecture on completely positive maps that could offer an alternative route to an area law via CP-map structure. The results have implications for numerical simulations by delineating conditions under which ground states are well-approximated by MPS and for understanding the interplay between entropy measures and state approximation in gapped 1D systems.

Abstract

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 completely positive maps 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.

An Area Law for One Dimensional Quantum Systems

TL;DR

The paper proves a rigorous area law for entanglement entropy in one-dimensional gapped quantum systems, deriving a bound S_max that scales exponentially with the correlation length. It introduces a boundary-operator framework (O_B,O_L,O_R) grounded in Lieb-Robinson bounds and a spectral gap, plus a bootstrap argument on block entropies that yields a finite entropy bound and connects to matrix product state representations. It also discusses why data-hiding and quantum expander constructions can complicate naive expectations, and presents a conjecture on completely positive maps that could offer an alternative route to an area law via CP-map structure. The results have implications for numerical simulations by delineating conditions under which ground states are well-approximated by MPS and for understanding the interplay between entropy measures and state approximation in gapped 1D systems.

Abstract

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 completely positive maps 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 and implies the ability to approximate the ground state by a matrix product state.

Paper Structure

This paper contains 5 sections, 5 theorems, 57 equations, 1 figure.

Table of Contents

  1. Proof of Main Theorem
  2. Matrix Product States
  3. Discussion and a Conjecture on Completely Positive Maps
  4. Conclusion
  5. Appendix: Approximating the Projection Operator

Key Result

Theorem 1

Consider a Hamiltonian satisfying the finite range and finite interaction strength conditions above. Suppose $H$ has a unique ground state with a gap $\Delta E$ to the first excited state. Then, for any $i$, where we define for some numerical constant $c_0$ of order unity, and where we define

Figures (1)

  • Figure 1: Illustration of an expander graph, with a particular choice of $\alpha,\beta,...$ marked. Note that each Greek index marks a site which is one site away from the previous index.

Theorems & Definitions (9)

  • Theorem
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Conjecture 1
  • Lemma 4
  • proof