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Embezzlement of entanglement, quantum fields, and the classification of von Neumann algebras

Lauritz van Luijk, Alexander Stottmeister, Reinhard F. Werner, Henrik Wilming

TL;DR

This work develops a comprehensive operational framework for embezzlement of entanglement within von Neumann algebras, linking the possibility and quality of embezzlement to Connes' classification of factors via the flow of weights. The authors define κ(ω) and related invariants κ_min(𝓜), κ_max(𝓜), revealing a sharp dichotomy: semifinite algebras admit no embezzlement (κ_min=κ_max=2), while type III factors with λ>0 admit embezzling states and, in particular, Type III$_1$ factors host universal embezzlers (κ_max equals the diameter of the state space, approaching 0). They show that the unique hyperfinite universal embezzler is the Araki–Woods factor 𝓡_∞, and they extend these results to infinite tensor products (ITPFI) and to quantum field theory, where local algebras are universal embezzlers in vacuum representations. The findings provide an operational interpretation of type III$_1$ in quantum information terms and offer a mechanism to understand maximal vacuum entanglement and Bell-inequality violations in relativistic QFT, with potential implications for energy-scale considerations and quantum gravity.

Abstract

We study the quantum information theoretic task of embezzlement of entanglement in the setting of von Neumann algebras. Given a shared entangled resource state, this task asks to produce arbitrary entangled states using local operations without communication while perturbing the resource arbitrarily little. We quantify the performance of a given resource state by the worst-case error. States for which the latter vanishes are 'embezzling states' as they allow to embezzle arbitrary entangled states with arbitrarily small error. The best and worst performance among all states defines two algebraic invariants for von Neumann algebras. The first invariant takes only two values. Either it vanishes and embezzling states exist, which can only happen in type III, or no state allows for nontrivial embezzlement. In the case of factors not of finite type I, the second invariant equals the diameter of the state space. This provides a quantitative operational interpretation of Connes' classification of type III factors within quantum information theory. Type III$_1$ factors are 'universal embezzlers' where every state is embezzling. Our findings have implications for relativistic quantum field theory, where type III algebras naturally appear. For instance, they explain the maximal violation of Bell inequalities in the vacuum. Our results follow from a one-to-one correspondence between embezzling states and invariant probability measures on the flow of weights. We also establish that universally embezzling ITPFI factors are of type III$_1$ by elementary arguments.

Embezzlement of entanglement, quantum fields, and the classification of von Neumann algebras

TL;DR

This work develops a comprehensive operational framework for embezzlement of entanglement within von Neumann algebras, linking the possibility and quality of embezzlement to Connes' classification of factors via the flow of weights. The authors define κ(ω) and related invariants κ_min(𝓜), κ_max(𝓜), revealing a sharp dichotomy: semifinite algebras admit no embezzlement (κ_min=κ_max=2), while type III factors with λ>0 admit embezzling states and, in particular, Type III factors host universal embezzlers (κ_max equals the diameter of the state space, approaching 0). They show that the unique hyperfinite universal embezzler is the Araki–Woods factor 𝓡_∞, and they extend these results to infinite tensor products (ITPFI) and to quantum field theory, where local algebras are universal embezzlers in vacuum representations. The findings provide an operational interpretation of type III in quantum information terms and offer a mechanism to understand maximal vacuum entanglement and Bell-inequality violations in relativistic QFT, with potential implications for energy-scale considerations and quantum gravity.

Abstract

We study the quantum information theoretic task of embezzlement of entanglement in the setting of von Neumann algebras. Given a shared entangled resource state, this task asks to produce arbitrary entangled states using local operations without communication while perturbing the resource arbitrarily little. We quantify the performance of a given resource state by the worst-case error. States for which the latter vanishes are 'embezzling states' as they allow to embezzle arbitrary entangled states with arbitrarily small error. The best and worst performance among all states defines two algebraic invariants for von Neumann algebras. The first invariant takes only two values. Either it vanishes and embezzling states exist, which can only happen in type III, or no state allows for nontrivial embezzlement. In the case of factors not of finite type I, the second invariant equals the diameter of the state space. This provides a quantitative operational interpretation of Connes' classification of type III factors within quantum information theory. Type III factors are 'universal embezzlers' where every state is embezzling. Our findings have implications for relativistic quantum field theory, where type III algebras naturally appear. For instance, they explain the maximal violation of Bell inequalities in the vacuum. Our results follow from a one-to-one correspondence between embezzling states and invariant probability measures on the flow of weights. We also establish that universally embezzling ITPFI factors are of type III by elementary arguments.
Paper Structure (27 sections, 82 theorems, 241 equations, 1 figure, 1 table)

This paper contains 27 sections, 82 theorems, 241 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Conclusion and outlook
  3. Acknowledgements
  4. Preliminaries
  5. Hilbert spaces, von Neumann algebras, and normal states
  6. Weights and modular theory
  7. Crossed products
  8. Spectral scales
  9. Embezzling states
  10. Equivalent notions of embezzlement
  11. Standard bipartite systems
  12. Spectral properties
  13. Embezzlement and the flow of weights
  14. The spectral state
  15. Semifinite von Neumann algebras
  16. ...and 12 more sections

Key Result

Theorem 2

For any bipartite system $({\mathcal{H}},{\mathcal{M}},{\mathcal{M}}')$, a state $\Omega\in{\mathcal{H}}$ is an embezzling state if and only if its induced states $\omega$ and $\omega'$ on ${\mathcal{M}}$ and ${\mathcal{M}}'$, respectively, are monopartite embezzling states.

Figures (1)

  • Figure 1: Spectral scale $\lambda_\omega(t)$ and distribution function $D_\omega(t)$ of a state $\omega = \mathop{\mathrm{Tr}}\nolimits (\rho\,{\,\cdot\,})$ on $M_n$ with density operator $\rho\in M_n$. The numbers $p_i$ are the eigenvalues of $\rho$, ordered increasingly, and $m_i\in{\mathbb N}$ denotes their multiplicity.

Theorems & Definitions (167)

  • Definition 1: Embezzling state
  • Theorem 2: cf. Thm. \ref{['thm:partite']}
  • Theorem 3: cf. Prop. \ref{['thm:spectral_charac']}
  • Theorem 4: cf. Thm. \ref{['thm:kappa']}
  • Theorem 5: cf. Thms. \ref{['thm:min']} and \ref{['thm:max']}
  • Theorem 6: cf. Thm. \ref{['thm:kappa_bound']}
  • Theorem 7: cf. Cor. \ref{['cor:itpfi-III1']}
  • Theorem 8: cf. Cor. \ref{['cor:itpfi-III1-mixed']}
  • Theorem 9: cf. Cor. \ref{['cor:exact-mbz']}
  • Theorem 10: cf. Prop. \ref{['prop:beta']}
  • ...and 157 more