Some Preliminary Notions
Before stating Choi's result, we give the definition of a completely positive map and fix some notation. Cn × n will denote the C*-algebra of n × n complex matrices. We will call A ∈ Cn × n positive, or symbolically, A ≥ 0, if A is Hermitian and the spectrum of A is nonnegative. (This condition is also called positive semidefinite.)
A linear map Φ : Cn × n → Cm × m is said to be a positive map if Φ(A) ≥ 0 for all A ≥ 0. In other words, a map Φ is positive if it preserves Hermiticity and the cone of positive elements.
Any linear map Φ induces another map
in a natural way: define
and extend by linearity. In matrix notation, a general element in
can be expressed as a k × k operator matrix:
and its image under the induced map is
Writing out the individual elements in the above matrix-of-matrices amounts to the natural identification of algebras
We say that Φ is k-positive if, considered as an element of Ckm×km, is a positive map, and Φ is called completely positive if Φ is k-positive for all k.
The transposition map is a standard example of a positive map that fails to be 2-positive. Let T denote this map on C 2 × 2. The following is a positive matrix in :
The image of this matrix under is
which is clearly not positive, having determinant -1.
Incidentally, a map Φ is said to be co-positive if the composition Φ T is positive. The transposition map itself is a co-positive map.
The above notions concerning positive maps extend naturally to maps between C*-algebras.
Read more about this topic: Choi's Theorem On Completely Positive Maps
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