Nielsen's theorem

Nielsen's theorem is a result in quantum information concerning transformations between bipartite states due to Michael Nielsen.^[1] It makes use of majorization.

Statement

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

This can be written more concisely as

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

Proof

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