Pure state entanglement and von Neumann algebras
arXiv:2409.17739 · doi:10.1007/s00220-025-05465-5
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 III 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 -finite measure spaces.
35+13+5 pages, 1 figure; v2: improved presentation, added references
References in corpus (10)
- An Algebra of Observables for de Sitter Space
- Gravity and the Crossed Product
- Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory
- Entanglement, Haag-duality and type properties of infinite quantum spin chains
- Complete Characterization of Entanglement Embezzlement
- On Haag Duality for Pure States of Quantum Spin Chain
- The Schmidt rank for the commuting operator framework
- Relativistic Quantum Fields Are Universal Entanglement Embezzlers
- Entanglement cost for infinite-dimensional physical systems
- Multipartite Embezzlement of Entanglement