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  transforms to another  using local operations and classical communication if and only if  is majorized by  where the  are the Schmidt coefficients of the respective state. This can be written more concisely as

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

Quantum information theory