Nielsen's theorem

Nielsen's theorem is a result in quantum information concerning transformations between bipartite states due to Michael Nielsen. It makes use of majorization.

Statement

A bipartite state $|\psi \rangle$ transforms to another $|\phi \rangle$ using local operations and classical communication if and only if $\lambda _{\psi }$ is majorized by $\lambda _{\phi }$ where the $\lambda _{i}$ are the Schmidt coefficients of the respective state.

This can be written more concisely as

$|\psi \rangle \rightarrow |\phi \rangle$ iff $\lambda _{\psi }\prec \lambda _{\phi }$.

Proof

The proof is detailed in the paper and will be added here at a later date.