Choi's theorem on completely positive maps

In mathematics, Choi's theorem on completely positive maps (after Man-Duen Choi) is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon–Nikodym" theorem for completely positive maps.

We observe that if

then E=E^* and E²=nE, so E=n⁻¹EE^* which is positive. Therefore C_Φ =(I_n ⊗ Φ)(E) is positive by the n-positivity of Φ.

This holds trivially.

This mainly involves chasing the different ways of looking at C^nm×nm:

Let the eigenvector decomposition of C_Φ be

where the vectors ${\displaystyle v_{i}}$ lie in C^nm . By assumption, each eigenvalue ${\displaystyle \lambda _{i}}$ is non-negative so we can absorb the eigenvalues in the eigenvectors and redefine ${\displaystyle v_{i}}$ so that

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