Choi's Theorem On Completely Positive Maps - Some Preliminary Notions

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

( I_k \otimes \Phi ) (M \otimes A) = M \otimes \Phi (A)

and extend by linearity. In matrix notation, a general element in

can be expressed as a k × k operator matrix:

\begin{bmatrix} A_{11} & \cdots & A_{1k} \\ \vdots & \ddots & \vdots \\ A_{k1} & \cdots & A_{kk} \end{bmatrix},

and its image under the induced map is

(I_k \otimes \Phi) (\begin{bmatrix} A_{11} & \cdots & A_{1k} \\ \vdots & \ddots & \vdots \\A_{k1} & \cdots & A_{kk} \end{bmatrix}) = \begin{bmatrix} \Phi (A_{11}) & \cdots & \Phi( A_{1k} ) \\ \vdots & \ddots & \vdots \\ \Phi (A_{k1}) & \cdots & \Phi( A_{kk} ) \end{bmatrix}.

Writing out the individual elements in the above matrix-of-matrices amounts to the natural identification of algebras

\mathbb{C}^{k\times k}\otimes\mathbb{C}^{m\times m}\cong\mathbb{C}^{km\times km}.

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 :

\begin{bmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}& \begin{pmatrix}0&1\\0&0\end{pmatrix}\\ \begin{pmatrix}0&0\\1&0\end{pmatrix}& \begin{pmatrix}0&0\\0&1\end{pmatrix} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ \end{bmatrix} .

The image of this matrix under is

\begin{bmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}^T& \begin{pmatrix}0&1\\0&0\end{pmatrix}^T\\ \begin{pmatrix}0&0\\1&0\end{pmatrix}^T& \begin{pmatrix}0&0\\0&1\end{pmatrix}^T \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} ,

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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