In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras.
This mainly involves chasing the different ways of looking at Cnm×nm: Let the eigenvector decomposition of CΦ be where the vectors
so that The vector space Cnm can be viewed as the direct sum
If Pk ∈ Cm × nm is projection onto the k-th copy of Cm, then Pk* ∈ Cnm×m is the inclusion of Cm as the k-th summand of the direct sum and Now if the operators Vi ∈ Cm×n are defined on the k-th standard basis vector ek of Cn by then Extending by linearity gives us for any A ∈ Cn×n.
Notice, given a completely positive Φ, its Kraus operators need not be unique.
For example, any "square root" factorization of the Choi matrix CΦ = B∗B gives a set of Kraus operators.
Let where bi*'s are the row vectors of B, then The corresponding Kraus operators can be obtained by exactly the same argument from the proof.
When the Kraus operators are obtained from the eigenvector decomposition of the Choi matrix, because the eigenvectors form an orthogonal set, the corresponding Kraus operators are also orthogonal in the Hilbert–Schmidt inner product.
This is not true in general for Kraus operators obtained from square root factorizations.
(Positive semidefinite matrices do not generally have a unique square-root factorizations.)
If two sets of Kraus operators {Ai}1nm and {Bi}1nm represent the same completely positive map Φ, then there exists a unitary operator matrix This can be viewed as a special case of the result relating two minimal Stinespring representations.
Alternatively, there is an isometry scalar matrix {uij}ij ∈ Cnm × nm such that This follows from the fact that for two square matrices M and N, M M* = N N* if and only if M = N U for some unitary U.
It follows immediately from Choi's theorem that Φ is completely copositive if and only if it is of the form Choi's technique can be used to obtain a similar result for a more general class of maps.
Since Hermitian matrices do not admit factorizations of the form B*B in general, the Kraus representation is no longer possible for a given Φ.