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

TL;DR

This paper explores the concept of embezzlement of entanglement within von Neumann algebras, linking quantum information tasks to the algebraic classification of these algebras, especially type III factors, with implications for quantum field theory.

Contribution

It introduces algebraic invariants for von Neumann algebras based on embezzlement performance, providing a novel operational interpretation of Connes' classification within quantum information theory.

Findings

Embezzling states exist only in type III von Neumann algebras.

Type III₁ factors are universal embezzlers, allowing all states to be embezzled.

The second invariant equals the diameter of the state space for non-finite type I factors.

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.