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Entanglement in von Neumann Algebraic Quantum Information Theory

Lauritz van Luijk

TL;DR

This work establishes a rigorous operational bridge between infinite quantum entanglement and the abstract type classification of von Neumann algebras. By modeling subsystems with von Neumann algebras and extending LOCC concepts to this setting, it shows a one-to-one correspondence between factor types (and subtypes) and families of entanglement properties, with Connes' type ${\mathrm {III}}$ linked to the smallest embezzlement error. Key insights include that type ${\mathrm{III}}_1$ systems are LU-transitive and universal embezzlers, enabling almost perfect state transformations via local unitaries, and that multipartite embezzling constructions exist in infinite settings. The framework connects concrete physical models (ground states in spin chains, QFT) to algebraic classifications, yielding testable predictions about how infinite entanglement manifests in different systems and guiding future exploration of Haag duality, flow of weights, and operational entanglement invariants.

Abstract

In quantum systems with infinitely many degrees of freedom, states can be infinitely entangled across a pair of subsystems, but are there different forms of infinite entanglement? To understand entanglement in such systems, we use a framework in which subsystems are described by von Neumann algebras on the full system's Hilbert space. Although this approach has been known for over 50 years, an operational justification has been missing so far. We resolve this by deriving the von Neumann algebraic description of subsystems from operational axioms. This raises the question of how physical properties of the subsystem relate to algebraic properties. Our main result shows a surprisingly strong connection: The type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with a family of operational entanglement properties. For instance, Connes' classification of type III factors can be formulated in terms of the smallest achievable error when "embezzling" entanglement from the system. Our findings promote the type classification from algebraic bookkeeping to a classification of infinite quantum systems based on the kind of infinite entanglement that they support.

Entanglement in von Neumann Algebraic Quantum Information Theory

TL;DR

This work establishes a rigorous operational bridge between infinite quantum entanglement and the abstract type classification of von Neumann algebras. By modeling subsystems with von Neumann algebras and extending LOCC concepts to this setting, it shows a one-to-one correspondence between factor types (and subtypes) and families of entanglement properties, with Connes' type linked to the smallest embezzlement error. Key insights include that type systems are LU-transitive and universal embezzlers, enabling almost perfect state transformations via local unitaries, and that multipartite embezzling constructions exist in infinite settings. The framework connects concrete physical models (ground states in spin chains, QFT) to algebraic classifications, yielding testable predictions about how infinite entanglement manifests in different systems and guiding future exploration of Haag duality, flow of weights, and operational entanglement invariants.

Abstract

In quantum systems with infinitely many degrees of freedom, states can be infinitely entangled across a pair of subsystems, but are there different forms of infinite entanglement? To understand entanglement in such systems, we use a framework in which subsystems are described by von Neumann algebras on the full system's Hilbert space. Although this approach has been known for over 50 years, an operational justification has been missing so far. We resolve this by deriving the von Neumann algebraic description of subsystems from operational axioms. This raises the question of how physical properties of the subsystem relate to algebraic properties. Our main result shows a surprisingly strong connection: The type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with a family of operational entanglement properties. For instance, Connes' classification of type III factors can be formulated in terms of the smallest achievable error when "embezzling" entanglement from the system. Our findings promote the type classification from algebraic bookkeeping to a classification of infinite quantum systems based on the kind of infinite entanglement that they support.

Paper Structure

This paper contains 68 sections, 117 theorems, 279 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Setup.
  4. Entanglement theory.
  5. Strong forms of infinite entanglement.
  6. Discussion and outlook
  7. Preliminaries on von Neumann algebras
  8. The basics
  9. Hilbert spaces and bounded operators
  10. Von Neumann algebras
  11. Classification of von Neumann algebras
  12. Weights and states
  13. Semifinite von Neumann algebras
  14. Lp-spaces
  15. Spectral scales and majorization
  16. ...and 53 more sections

Key Result

Theorem 1

The axioms above hold if and only if there is a von Neumann algebra ${\mathcal{M}}$ on ${\mathcal{H}}$ such that the agent's operations are exactly the ${\mathcal{M}}$-inner ones.

Figures (4)

  • Figure 1: The classification of factors into types ${\mathrm{I}}$, ${\mathrm{II}}$, ${\mathrm{III}}$, and their respective subtypes (see \ref{['sec:vN-prelims']} for details).
  • Figure 2: Bipartition of an infinite quantum spin system into a cone and its complement.
  • Figure 3: Penrose diagram of Minkowski spacetime with one spatial dimension, partitioned into a pair ${\mathcal{O}}_A,{\mathcal{O}}_B$ of causally separated regions with ${\mathcal{O}}_A={\mathcal{O}}_B'$.
  • Figure 4: Three parties sharing infinitely many entangled states.

Theorems & Definitions (218)

  • Theorem 1: See \ref{['sec:axioms']}
  • Theorem 2: see \ref{['sec:haag']}
  • Theorem 3: see \ref{['sec:mbz', 'sec:locc-types']}
  • Theorem 4: see \ref{['sec:universal-mbz']}
  • Lemma 3.1: takesaki1
  • Example 3.2
  • Theorem 3.3: hiai_majorization_1987
  • Example 3.4
  • Theorem 3.5: Haagerup haagerup_standard_1975
  • Example 3.6
  • ...and 208 more