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 [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.
References
- ↑ Nielsen, M. A. (1999-07-12). "Conditions for a Class of Entanglement Transformations" (in en). Physical Review Letters 83 (2): 436–439. doi:10.1103/PhysRevLett.83.436. ISSN 0031-9007. https://link.aps.org/doi/10.1103/PhysRevLett.83.436.
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