Pure state entanglement and von Neumann algebras

TL;DR

This paper extends the theory of local operations and classical communication (LOCC) to bipartite quantum systems modeled by von Neumann algebras, revealing how algebraic types influence entanglement properties and state transformations.

Contribution

It generalizes Nielsen's Theorem to arbitrary von Neumann factors and links algebraic types with operational entanglement characteristics.

Findings

Type III factors allow LOCC transitions of arbitrary precision between pure states.

All states have infinite single-shot entanglement if local factors are not of type I.

Pure state entanglement monotones extend to semifinite factors.

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$_{1}$ 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 $\sigma$-finite measure spaces.