Exact entanglement embezzlement is a self-test: any protocol for a fixed target state is unitarily equivalent and generates a unique AFD type IIIλ factor whose λ is determined by the Schmidt coefficients.
van Luijk, A
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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.
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The f_0-divergence defined via Jordan decomposition integrals coincides with Araki's relative entropy on arbitrary von Neumann algebras, extending Frenkel's finite-dimensional formula.
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The paper organizes important open questions in quantum gravity and quantum information into four themes without presenting new results or derivations.
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Self-testing of exact entanglement embezzlement
Exact entanglement embezzlement is a self-test: any protocol for a fixed target state is unitarily equivalent and generates a unique AFD type IIIλ factor whose λ is determined by the Schmidt coefficients.
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Integral representations of $f$-divergences for general von Neumann algebras
The f_0-divergence defined via Jordan decomposition integrals coincides with Araki's relative entropy on arbitrary von Neumann algebras, extending Frenkel's finite-dimensional formula.
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Modular Lower Bounds on Reeh-Schlieder State Preparation
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Rethinking quantum information in gravity and fields
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