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 C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon-Nikodym" theorem for completely positive maps.

Statement

Choi's theorem. Let be a linear map. The following are equivalent:

Proof

(i) implies (ii)

We observe that if
then E=E* and E2=nE, so E=n−1EE* which is positive. Therefore CΦ = is positive by the n-positivity of Φ.

(iii) implies (i)

This holds trivially.

(ii) implies (iii)

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 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.

Consequences

Kraus operators

In the context of quantum information theory, the operators are called the Kraus operators 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 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.
If two sets of Kraus operators 1nm and 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 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.

Completely copositive maps

It follows immediately from Choi's theorem that Φ is completely copositive if and only if it is of the form

Hermitian-preserving maps

Choi's technique can be used to obtain a similar result for a more general class of maps. Φ is said to be Hermitian-preserving if A is Hermitian implies Φ is also Hermitian. One can show Φ is Hermitian-preserving if and only if it is of the form
where λi are real numbers, the eigenvalues of CΦ, and each Vi corresponds to an eigenvector of CΦ. Unlike the completely positive case, CΦ may fail to be positive. 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.