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Communications in Mathematical Physics

Pure State Entanglement and von Neumann Algebras

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

Published Oct 30, 2025• 3 citations
Publisher

Abstract

Abstract We develop the theory of local operations and classical communication (LOCC) for bipartite quantum systems represented by commuting von Neumann algebras. Our central result is the extension of Nielsen’s Theorem, stating that the LOCC ordering of bipartite pure states is equivalent to the majorization of their restrictions to arbitrary factors. As a consequence, we find that in bipartite system modeled by commuting factors in Haag duality, (a) all states have infinite single-shot entanglement if and only if the local factors are not of type I, (b) type III factors are characterized by LOCC transitions of arbitrary precision between any two pure states, and c) the latter holds even without classical communication for type $$\hbox {III}_{1}$$ III 1 factors. In the case of semifinite factors, the usual construction of pure state entanglement monotones carries over. Together with recent work on embezzlement of entanglement, this gives a one-to-one correspondence between the classification of factors into types and subtypes and operational entanglement properties. In the appendix, we provide a self-contained treatment of majorization on semifinite von Neumann algebras and $$\sigma $$ σ -finite measure spaces.

DOI
10.1007/s00220-025-05465-5

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Abstract We develop the theory of local operations and classical communication (LOCC) for bipartite quantum systems represented by commuting von Neumann algebras. Our central result is the extension of Nielsen’s Theorem, stating that the LOCC ordering of bipartite pure states is equivalent to the majorization of their restrictions to arbitrary factors. As a consequence, we find that in bipartite system modeled by commuting factors in Haag duality, (a) all states have infinite single-shot entanglement if and only if the local factors are not of type I, (b) type III factors are characterized by LOCC transitions of arbitrary precision between any two pure states, and c) the latter holds even without classical communication for type III1\hbox {III}_{1}III1​ III 1 factors. In the case of semifinite factors, the usual construction of pure state entanglement monotones carries over. Together with recent work on embezzlement of entanglement, this gives a one-to-one correspondence between the classification of factors into types and subtypes and operational entanglement properties. In the appendix, we provide a self-contained treatment of majorization on semifinite von Neumann algebras and σ\sigma σ σ -finite measure spaces.
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 answer this, we provide a rigorous framework for studying information-theoretic properties in infinite systems, formulated in terms of von Neumann algebras. In this framework, we show that there are operational tasks that can distinguish between different forms of infinite entanglement. We find that the type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with operational entanglement properties. For instance, the Connes classification of type III factors corresponds to the system’s capability at the operational task of “embezzling” entanglement. Our findings promote the type classification from algebraic bookkeeping to a classification of infinite quantum systems based on their operational entanglement properties.
Abstract We advance on the conversion of bipartite quantum states via local operations and classical communication (LOCC) for infinite-dimensional systems. We introduce δ-LOCC convertibility based on the observation that any pure state can be approximated by a state with finite-support Schmidt coefficients. We show that δ-LOCC convertibility of bipartite states is fully characterized by a majorization relation between the sequences of squared Schmidt coefficients, providing a novel extension of Nielsen’s theorem for infinite-dimensional systems. Hence, our definition is equivalent to the one of ε-LOCC convertibility (Owari et al 2008 Quantum Inf. Comput. 8 0030), but deals with states having finitely supported sequences of Schmidt coefficients. Additionally, we discuss the notions of optimal common resource and optimal common product in this scenario. The optimal common product always exists, whereas the optimal common resource depends on the existence of a common resource. This highlights a distinction between the resource-theoretic aspects of finite versus infinite-dimensional systems. Our results rely on the order-theoretic properties of majorization for infinite sequences, applicable beyond the LOCC convertibility problem.
We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.
Nielsen characterized in full those 2-party quantum protocols of local operations and classical communication that transform, with probability one, a pure global initial state into a pure global final state. The present work considers the generalization of Nielsen's characterization to n-party protocols. It presents a sweeping generalization of the only if part of Nielsen's result. The result presented here pertains also to protocols that do not generate a final state for sure, it considers arbitrary mixed initial states instead of pure states and n-party protocols for an arbitrary n. In this very general setting, local operations and classical communication can never decrease the expected spectra of the local mixed states in the majorization ordering. In other terms, the local states can only become purer (weakly) in expectation. The proof also provides an improvement on Nielsen's. The if part of Nielsen's characterization does not generalize. This is shown by studying the entanglement of three qubits. It is shown that one can find pure states of a system of three qubits that are not equivalent under unitary local operations but define local mixed states on all subparts of the system that have the same spectra. Neither equivalence of pure states under local unitary operations nor accessibility under LOCC operations among a system of three qubits can be characterized by properties of the spectra of the local mixed states.
We study conditions for the deterministic transformation $\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath{\rightarrow}\ensuremath{\mid}\ensuremath{\phi}⟩$ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen's majorization theorem provides necessary and sufficient conditions. For the general case, we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a deterministic map produced by another separable operation, if one excludes the case where the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$. The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$, we show that the Kraus operators of the separable transformation restricted to the supports of $\ensuremath{\mid}\ensuremath{\psi}⟩$ on the factor spaces are proportional to unitaries. When that proportionality holds and the factor spaces have equal dimension, we find conditions for the deterministic transformation of a collection of several full Schmidt rank pure states $\ensuremath{\mid}{\ensuremath{\psi}}_{j}⟩$ to pure states $\ensuremath{\mid}{\ensuremath{\phi}}_{j}⟩$.
We show that the possible ensembles produced when a separable operation acts on a single pure bipartite entangled state are completely characterized by a majorization condition, a collection of inequalities for Schmidt coefficients, which is identical to that already known for the particular case of local operations and classical communication (LOCC). As a consequence, various known results for LOCC, including some involving monotonicity of entanglement, can be extended to the class of all separable operations.