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.