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 $|\psi ⟩$ transforms to another $|\varphi ⟩$ using local operations and classical communication if and only if ${\lambda }_{\psi }$ is majorized by ${\lambda }_{\varphi }$ where the ${\lambda }_i$ are the Schmidt coefficients of the respective state.

This can be written more concisely as

$|\psi ⟩\to |\varphi ⟩$ iff ${\lambda }_{\psi }\prec {\lambda }_{\varphi }$.

Proof

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