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Entanglement in quantum spin chains is strictly finite at any temperature

Ainesh Bakshi, Soonwon Choi, Saúl Pilatowsky-Cameo

Abstract

Entanglement is the hallmark of quantum physics, yet its characterization in interacting many-body systems at thermal equilibrium remains one of the most important challenges in quantum statistical physics. We prove that the Gibbs state of any quantum spin chain can be exactly decomposed into a mixture of matrix product states with a bond dimension that is independent of the system size, at any finite temperature. As a consequence, the Schmidt number, arguably the most stringent measure of bipartite entanglement, is strictly finite for thermal states, even in the thermodynamic limit. Our decomposition is explicit and is accompanied by an efficient classical algorithm to sample the resulting matrix product states.

Entanglement in quantum spin chains is strictly finite at any temperature

Abstract

Entanglement is the hallmark of quantum physics, yet its characterization in interacting many-body systems at thermal equilibrium remains one of the most important challenges in quantum statistical physics. We prove that the Gibbs state of any quantum spin chain can be exactly decomposed into a mixture of matrix product states with a bond dimension that is independent of the system size, at any finite temperature. As a consequence, the Schmidt number, arguably the most stringent measure of bipartite entanglement, is strictly finite for thermal states, even in the thermodynamic limit. Our decomposition is explicit and is accompanied by an efficient classical algorithm to sample the resulting matrix product states.
Paper Structure (38 sections, 18 theorems, 109 equations, 1 figure, 1 table)

This paper contains 38 sections, 18 theorems, 109 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Thermal states are exact mixtures of matrix product states
  4. Bound on thermal entanglement in 1D
  5. Efficient classical simulation algorithm
  6. Technical overview
  7. Entanglement bulk decomposition.
  8. Mixture of matrix product states
  9. Ladder of local operators.
  10. Separability from quasilocality.
  11. Thermal state as a mixture of MPSs.
  12. Proof sketch of the entanglement bulk decomposition
  13. Araki expansional.
  14. Locality in uniform hypergraphs.
  15. Bound on path sums.
  16. ...and 23 more sections

Key Result

Theorem 2.1

(1D Thermal states are exact mixtures of MPSs) In any geometrically $\mathfrak{K}$-local Hamiltonian on a spin chain of length $n$ and at any inverse temperature $0 \leqslant \beta<\infty$, the Gibbs state $g_\beta$ admits the following decomposition (see Fig. fig:01): where each $M_i$ is supported between qubits $i$ and $i+m-1$, for $m={\exp\lparen*\rparen{c\tilde{\beta}}}$ with $c \leqslant \ex

Figures (1)

  • Figure 1: Tensor network diagram of the decomposition of \ref{['th:GibbsStatesareMixturesOfMPS']}, where the Gibbs state $g_\beta$ is decomposed as a ladder of local operators acting on a separable state $\sigma$. Each vertical line represents a qubit, and each box represents an operator supported within. Vertically stacking boxes represents tensor contraction or equivalently matrix multiplication.

Theorems & Definitions (39)

  • Theorem 2.1
  • Corollary 2.2: Entanglement is strictly finite
  • Theorem 2.3: Efficient classical simulation algorithm
  • Definition 3.1: Valid growth set (informal, see \ref{['def:growth-sets']})
  • Definition 5.1: Pauli matrices, Pauli strings, and stabilizer product states
  • Definition 5.2: Quasilocal operators and quasilocal perturbations of the identity
  • Definition 5.3: Separable operator
  • Definition 5.4: Rényi entanglement of formation
  • Definition 5.5: Schmidt number
  • Proposition 6.1: Quasilocal perturbations of the identity are separable
  • ...and 29 more