Choi's theorem on completely positive maps

In mathematics, Choi's theorem on completely positive maps 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. Choi's theorem. Let be a linear map. The following are equivalent: (i) Φ is n-positive (i.e. is positive whenever is positive). (ii) The matrix with operator entries is positive, where is the matrix with 1 in the ij-th entry and 0s elsewhere. (The matrix CΦ is sometimes called the Choi matrix of Φ.) (iii) Φ is completely positive. We observe that if then E=E* and E2=nE, so E=n−1EE* which is positive. Therefore CΦ =(In ⊗ Φ)(E) is positive by the n-positivity of Φ. This holds trivially. This mainly involves chasing the different ways of looking at Cnm×nm: Let the eigenvector decomposition of CΦ be where the vectors lie in Cnm . By assumption, each eigenvalue is non-negative so we can absorb the eigenvalues in the eigenvectors and redefine so that The vector space Cnm can be viewed as the direct sum compatibly with the above identification and the standard basis of Cn. 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. Any map of this form is manifestly completely positive: the map is completely positive, and the sum (across ) of completely positive operators is again completely positive. Thus is completely positive, the desired result. The above is essentially Choi's original proof. Alternative proofs have also been known. In the context of quantum information theory, the operators {Vi} are called the Kraus operators (after Karl Kraus) of Φ. 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.