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 [math]\displaystyle{ |\psi\rangle }[/math] transforms to another [math]\displaystyle{ | \phi \rangle }[/math] using local operations and classical communication if and only if [math]\displaystyle{ \lambda_{\psi} }[/math] is majorized by [math]\displaystyle{ \lambda_{\phi} }[/math] where the [math]\displaystyle{ \lambda_i }[/math] are the Schmidt coefficients of the respective state.

This can be written more concisely as

[math]\displaystyle{ |\psi \rangle \rightarrow |\phi\rangle }[/math] iff [math]\displaystyle{ \lambda_{\psi} \prec \lambda_{\phi} }[/math].

Proof

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