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Spatial Entanglement Sudden Death in Spin Chains at All Temperatures

Samuel O. Scalet

Abstract

We prove a finite entanglement length for the Gibbs state of any local Hamiltonian on a spin chain at any finite temperature: After removing an interval of size at least equal to the entanglement length, the remaining left and right half-chains are in a separable state.

Spatial Entanglement Sudden Death in Spin Chains at All Temperatures

Abstract

We prove a finite entanglement length for the Gibbs state of any local Hamiltonian on a spin chain at any finite temperature: After removing an interval of size at least equal to the entanglement length, the remaining left and right half-chains are in a separable state.
Paper Structure (9 sections, 12 theorems, 59 equations, 2 figures)

This paper contains 9 sections, 12 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Related work
  4. Discussion and outlook
  5. Preliminaries
  6. Separable states
  7. 1D Gibbs states
  8. Proof of Theorem \ref{['thm:mainTec']}
  9. The thermodynamic limit

Key Result

Theorem 1

For a local Hamiltonian on a spin chain, there exists a constant $\ell\in\mathbb{N}$ that only depends on the temperature and locality, but not on the system size, such that the following holds. For any tripartite interval $ABC$ with and $\rho^{ABC}=\exp(-\beta H_{ABC})/Z_{ABC}$ the Gibbs state on the interval, we have that is separable between $A$ and $C$.

Figures (2)

  • Figure 1: Graphical representation of the regions $ABC$.
  • Figure 2: Illustration of the decomposition used in our proof. In addition to the separable contribution $\Gamma(k_0)$, the superexponentially decaying remainder terms $\Delta_k$ are added to the identity contribution $\gamma(k)\mathds{1}_{AC}$ (not depicted), leaving it separable.

Theorems & Definitions (26)

  • Theorem 1: Informal version of Theorem \ref{['thm:mainTec']}
  • Corollary 1: Informal version of Corollary \ref{['cor:KMS']}
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2: Gurvits_2002
  • Lemma 3
  • proof
  • Lemma 4: bluhm2022
  • Lemma 5
  • ...and 16 more