Details
| Original language | English |
|---|---|
| Article number | 296 |
| Journal | Communications in Mathematical Physics |
| Volume | 406 |
| Issue number | 12 |
| Publication status | Published - 30 Oct 2025 |
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 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-finite measure spaces.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematics(all)
- Mathematical Physics
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In: Communications in Mathematical Physics, Vol. 406, No. 12, 296, 30.10.2025.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Pure State Entanglement and von Neumann Algebras
AU - van Luijk, Lauritz
AU - Stottmeister, Alexander
AU - Werner, Reinhard F.
AU - Wilming, Henrik
N1 - Publisher Copyright: © The Author(s) 2025.
PY - 2025/10/30
Y1 - 2025/10/30
N2 - 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 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-finite measure spaces.
AB - 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 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-finite measure spaces.
UR - http://www.scopus.com/inward/record.url?scp=105020391491&partnerID=8YFLogxK
U2 - 10.1007/s00220-025-05465-5
DO - 10.1007/s00220-025-05465-5
M3 - Article
AN - SCOPUS:105020391491
VL - 406
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 12
M1 - 296
ER -