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The Large-Scale Structure of Entanglement in Quantum Many-body Systems

Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming

TL;DR

The work develops an operator-algebraic framework to study entanglement in the thermodynamic limit of quantum many-body systems, linking entanglement properties to the type of von Neumann factors arising from local observable algebras. Haag duality serves as a key structural principle, enabling a classification of bipartite entanglement via factor types ${\mathrm{I}}, {\mathrm{II}}, {\mathrm{III}}$ and their subtypes, with clear operational meanings such as finite one-shot entanglement, embezzlement, and LOCC interconvertibility. In $D\ge 2$ dimensions, every gapped phase contains models exhibiting the strongest large-scale entanglement (type ${\mathrm{III}}_1$), while topological order imposes lower bounds on entanglement strength and relates to fusion-category data; 1D results separate gapped from critical behavior via type ${\mathrm{I}}$ vs ${\mathrm{III}}_1$ sectors. The paper also connects finite-size behavior to thermodynamic-limit predictions, discusses embezzlement as a robust feature, and outlines open questions spanning symmetry breaking, multipartite entanglement, and potential links to gravity-inspired frameworks.

Abstract

We show that the thermodynamic limit of a many-body system can reveal entanglement properties that are hard to detect in finite-size systems -- similar to how phase transitions only sharply emerge in the thermodynamic limit. The resulting operational entanglement properties are in one-to-one correspondence with abstract properties of the local observable algebras that emerge in the thermodynamic limit. These properties are insensitive to finite perturbations and hence describe the $\textit{large-scale structure of entanglement}$ of many-body systems. We formulate and discuss the emerging structures and open questions, both for gapped and gapless many-body systems. In particular, we show that every gapped phase of matter, even the trivial one, in $D\geq 2$ dimensions contains models with the strongest possible bipartite large-scale entanglement. Conversely, we conjecture the existence of topological phases of matter, where all representatives have the strongest form of entanglement.

The Large-Scale Structure of Entanglement in Quantum Many-body Systems

TL;DR

The work develops an operator-algebraic framework to study entanglement in the thermodynamic limit of quantum many-body systems, linking entanglement properties to the type of von Neumann factors arising from local observable algebras. Haag duality serves as a key structural principle, enabling a classification of bipartite entanglement via factor types and their subtypes, with clear operational meanings such as finite one-shot entanglement, embezzlement, and LOCC interconvertibility. In dimensions, every gapped phase contains models exhibiting the strongest large-scale entanglement (type ), while topological order imposes lower bounds on entanglement strength and relates to fusion-category data; 1D results separate gapped from critical behavior via type vs sectors. The paper also connects finite-size behavior to thermodynamic-limit predictions, discusses embezzlement as a robust feature, and outlines open questions spanning symmetry breaking, multipartite entanglement, and potential links to gravity-inspired frameworks.

Abstract

We show that the thermodynamic limit of a many-body system can reveal entanglement properties that are hard to detect in finite-size systems -- similar to how phase transitions only sharply emerge in the thermodynamic limit. The resulting operational entanglement properties are in one-to-one correspondence with abstract properties of the local observable algebras that emerge in the thermodynamic limit. These properties are insensitive to finite perturbations and hence describe the of many-body systems. We formulate and discuss the emerging structures and open questions, both for gapped and gapless many-body systems. In particular, we show that every gapped phase of matter, even the trivial one, in dimensions contains models with the strongest possible bipartite large-scale entanglement. Conversely, we conjecture the existence of topological phases of matter, where all representatives have the strongest form of entanglement.

Paper Structure

This paper contains 14 sections, 8 theorems, 28 equations, 3 figures, 1 table.

Table of Contents

  1. Local operations and ground state sectors of many-body systems
  2. Bipartite entanglement and von Neumann type
  3. Stability to perturbations
  4. Finite systems
  5. Ground states in $D=1$
  6. Gapped ground states in $D\ge 2$
  7. Long-range vs large-scale entanglement
  8. Outlook and discussion
  9. Infinite tensor products
  10. Details of the construction in $D\geq 2$
  11. The local observable algebras
  12. Stability of approximately finite-dimensional factors
  13. Proof of \ref{['prop:stable-type']} and \ref{['thm:stable-bipartite']}
  14. Proof of \ref{['prop:distillation']}

Key Result

Theorem 2

For a fixed partition of $\Gamma$ into regions $A_1,\ldots A_N\subset\Gamma$, all entanglement properties of the ground state sector ${\mathcal{H}}$ are encoded in the collection of commuting von Neumann algebras ${\mathcal{M}}_{A_1},\ldots {\mathcal{M}}_{A_N}$.

Figures (3)

  • Figure 1: Illustration of \ref{['prop:stable-type']} and \ref{['thm:stable-bipartite']}: Modifying a bipartition into two (properly) infinite regions by a finite amount does not change the bipartite large-scale entanglement properties listed in \ref{['tab:types']}.
  • Figure 2: The basic construction. Each circle corresponds to a lattice site $v\in \mathcal{V}$ in the vertex set $\mathcal{V}$ of a graph $\mathcal{G} = (\mathcal{V},\mathcal{E})$ with constant degree $d$ (here, $d=4$). Each site $v$ is subdivided into $d$ virtual spins $\widetilde{v}$ of dimension $m\geq 3$. Each edge $e\in\mathcal{E}$ is associated with a neighboring pair of virtual spins and a copy of a canonical purification $|\psi\rangle\in {\mathbb C}^m\otimes {\mathbb C}^m$ of a density matrix $\rho$ is placed on each such edge. By construction, the reduced state of every site is thus given by $\rho^{\otimes d}$. When the lattice is subdivided into two infinite parts $A$ and $B$, the two parts are only entangled through the infinite tensor product $|\psi\rangle^{\otimes \infty}$ along the boundary, determining the type of local factors ${\mathcal{M}}_A,{\mathcal{M}}_B$.
  • Figure 3: Illustration of ITPFI factors: Two half-infinite spin chains described by an infinite tensor product of bipartite entangled pure states $|\Psi_i\rangle$ (with marginals $\rho_{A,i}$ and $\rho_{B,i}$, respectively) give rise to a (infinite-dimensional) type ${\mathrm{I}}$ factor ${\mathcal{B}}({\mathcal{H}})$ with commuting subfactors ${\mathcal{M}}_A$ and ${\mathcal{M}}_B$ described by the ITPFI factors ${\mathcal{M}}_A = \mathrm{ITPFI}(\{\rho_{A,i}\})$ and ${\mathcal{M}}_B = \mathrm{ITPFI}(\{\rho_{B,i}\})$ which fulfill Haag duality, ${\mathcal{M}}_A'= {\mathcal{M}}_B$.

Theorems & Definitions (13)

  • Definition 1: keyl_infinitely_2003
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Proposition 5
  • Proposition 6: van_luijk_critical_2024van_luijk_multipartite_2024
  • Corollary 7
  • proof : Proof of \ref{['prop:stable-type']}
  • Lemma 8
  • proof
  • ...and 3 more