# State convertibility in the von Neumann algebra framework

## TL;DR

This paper extends the fundamental theorem of quantum state convertibility to bipartite systems modeled by von Neumann algebras, introducing new notions of LOCC and connecting them to algebraic concepts like singular numbers and majorisation.

## Contribution

It generalizes Nielsen's theorem to von Neumann algebra frameworks, defines appropriate LOCC operations, and introduces an entanglement monotone based on singular value distributions.

## Key findings

- Established a von Neumann algebra version of state convertibility theorem
- Connected approximate convertibility to singular numbers and majorisation
- Identified entropy of singular value distribution as an entanglement monotone

## Abstract

We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen's theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. As an application of our result in the setting of $II_1$-factors, we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of Vidal, thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we show that trace vectors play the role of maximally entangled states for general $II_1$-factors. Examples are drawn from infinite spin chains, quasi-free representations of the CAR, and discretised versions of the CCR.