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arXiv:2406.11747 · Nature Physics

Critical fermions are universal embezzlers

Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming

Published May 27, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract Universal embezzlers are bipartite quantum systems from which any entangled state may be extracted to arbitrary precision using local operations while perturbing the system arbitrarily little. Here we show that a universal embezzler can be created by bipartitioning any local, translation-invariant, critical free-fermionic many-body system on a one-dimensional lattice. The same property holds for locally interacting spin chains that are dual to the critical fermionic models by the Jordan–Wigner transformation. Furthermore, for any finite error and any targeted entangled state, a finite length of the chain is sufficient to embezzle said state within the given error. Hence, universal embezzlement is not restricted to the thermodynamic limit. As well as establishing the ubiquity of universal embezzlers in many-body physics, on a technical level, our main result establishes that the half-chain observable algebras associated with ground-state sectors of the given models are type III1 factors.

DOI
10.1038/s41567-025-02921-w

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Abstract Universal embezzlers are bipartite quantum systems from which any entangled state may be extracted to arbitrary precision using local operations while perturbing the system arbitrarily little. Here we show that a universal embezzler can be created by bipartitioning any local, translation-invariant, critical free-fermionic many-body system on a one-dimensional lattice. The same property holds for locally interacting spin chains that are dual to the critical fermionic models by the Jordan–Wigner transformation. Furthermore, for any finite error and any targeted entangled state, a finite length of the chain is sufficient to embezzle said state within the given error. Hence, universal embezzlement is not restricted to the thermodynamic limit. As well as establishing the ubiquity of universal embezzlers in many-body physics, on a technical level, our main result establishes that the half-chain observable algebras associated with ground-state sectors of the given models are type III1 factors.
Embezzlement of entanglement allows to extract arbitrary entangled states from a suitable embezzling state using only local operations while perturbing the resource state arbitrarily little. A natural family of embezzling states is given by ground states of non-interacting, critical fermions in one spatial dimension. This raises the question of whether the embezzlement operations can be restricted to Gaussian operations whenever one only wishes to extract Gaussian entangled states. We show that this is indeed the case and prove that the embezzling property is in fact a generic property of fermionic Gaussian states. Our results provide a fine-grained understanding of embezzlement of entanglement for fermionic Gaussian states in the finite-size regime and thereby bridge finite-size systems to abstract characterizations based on the classification of von Neumann algebras. To prove our results, we establish novel bounds relating the distance of covariances to the trace-distance of Gaussian states, which may be of independent interest.
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 III1_11​ 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 III1_11​ by elementary arguments.
We present an explicit construction of a universal embezzlement protocol in the C*-algebraic model of quantum information, that is equivalent to the commuting operator model. Our protocol enables exact embezzlement of arbitrary bipartite pure states using a single, fixed catalyst state. Unlike prior constructions that achieve only approximate embezzlement or require state-dependent catalysts, our approach is both exact and state-independent. The construction is explicit, based on simple *-automorphisms acting locally on infinite tensor products of CAR algebras with the underlying idea of the Hilbert hotel. In the dense-state case, the protocol naturally recovers the Type III_1 factor via the GNS construction, consistent with recent classification results. We further extend the construction to allow exact embezzlement of all states, at the cost of working with a non-separable C*-algebra. Despite the increase in algebraic size, the operational structure remains simple and localized. This offers a conceptually intuitive model for universal entanglement embezzlement in infinite-dimensional settings.
We construct translation-invariant pure quasifree fermionic states on the one-dimensional lattice whose half-chain von Neumann algebras are hyperfinite factors of type II∞\mathrm{II}_\inftyII∞​ or IIIλ\mathrm{III}_λIIIλ​ for any 0<λ≤10<λ\leq 10<λ≤1. Via the Jordan-Wigner-Araki transformation, these yield translation-invariant pure spin-chain states with the same half-chain factors, demonstrating that such factors need not be of type I or III. Each state is the unique ground state of a gapless, translation-invariant parent Hamiltonian with superpolynomially decaying interactions. For any fixed factor type, the interval entanglement entropy can exhibit widely varying growth rates, from arbitrarily slow divergence to nearly power-law growth, showing that entropy scaling does not determine the factor type. In contrast, we prove that gauge-invariant quadratic Hamiltonians with finitely many flavours and exponentially decaying hopping exhibit a dichotomy: their ground states either satisfy an area law with a type I half-chain factor, or have logarithmic entropy growth with a type III1\mathrm{III}_1III1​ factor. The counterexample was discovered with AI assistance.
Embezzlement of entanglement refers to the task of extracting entanglement from an entanglement resource via local operations and without communication while perturbing the resource arbitrarily little. Recently, the existence of embezzling states of bipartite systems of type III von Neumann algebras was shown. However, both the multipartite case and the precise relation between embezzling states and the notion of embezzling families, as originally defined by van Dam and Hayden, was left open. Here, we show that finite-dimensional approximations of multipartite embezzling states form multipartite embezzling families. In contrast, not every embezzling family converges to an embezzling state. We identify an additional consistency condition that ensures that an embezzling family converges to an embezzling state. This criterion distinguishes the embezzling family of van Dam and Hayden from the one by Leung, Toner, and Watrous. The latter generalizes to the multipartite setting. By taking a limit, we obtain a multipartite system of commuting type III 1 factors on which every state is an embezzling state. We discuss our results in the context of quantum field theory and quantum many-body physics. As open problems, we ask whether vacua of relativistic quantum fields in more than two spacetime dimensions are multipartite embezzling states and whether multipartite embezzlement allows for an operator-algebraic characterization.
Embezzlement of entanglement refers to the counterintuitive possibility of extracting entangled quantum states from a reference state of an auxiliary system (the "embezzler") via local quantum operations while hardly perturbing the latter. We uncover a deep connection between the operational task of embezzling entanglement and the mathematical classification of von Neumann algebras. Our result implies that relativistic quantum fields are universal embezzlers: any entangled state of any dimension can be embezzled from them with arbitrary precision. This provides an operational characterization of the infinite amount of entanglement present in the vacuum state of relativistic quantum field theories.