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Nielsen's theorem

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

This can be written more concisely as

| ψ ⟩ → | ϕ ⟩ {\displaystyle |\psi \rangle \rightarrow |\phi \rangle } iff λ ψ ≺ λ ϕ {\displaystyle \lambda _{\psi }\prec \lambda _{\phi }} .

Proof

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

References

  1. Nielsen, M. A. (1999-07-12). "Conditions for a Class of Entanglement Transformations". Physical Review Letters. 83 (2): 436–439. arXiv:quant-ph/9811053. doi:10.1103/PhysRevLett.83.436. ISSN 0031-9007.
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