Completely positive map

Definition

Let A and B be C*-algebras. A linear map \phi: A\to B is called a positive map if \phi maps positive elements to positive elements: a\geq 0 \implies \phi(a)\geq 0.

Any linear map \phi:A\to B induces another map

:\textrm{id} \otimes \phi : \mathbb{C}^{k \times k} \otimes A \to \mathbb{C}^{k \times k} \otimes B

in a natural way. If \mathbb{C}^{k\times k}\otimes A is identified with the C*-algebra A^{k\times k} of k\times k-matrices with entries in A, then \textrm{id}\otimes\phi acts as : \begin{pmatrix} a_{11} & \cdots & a_{1k} \ \vdots & \ddots & \vdots \ a_{k1} & \cdots & a_{kk} \end{pmatrix} \mapsto \begin{pmatrix} \phi(a_{11}) & \cdots & \phi(a_{1k}) \ \vdots & \ddots & \vdots \ \phi(a_{k1}) & \cdots & \phi(a_{kk}) \end{pmatrix}.

We then say \phi is k-positive if \textrm{id}_{\mathbb{C}^{k\times k}} \otimes \phi is a positive map and completely positive if \phi is k-positive for all k.

Properties

• Positive maps are monotone, i.e. a_1\leq a_2\implies \phi(a_1)\leq\phi(a_2) for all self-adjoint elements a_1,a_2\in A_{sa}.
• Since -|a|_A 1_A \leq a \leq |a|A 1_A for all self-adjoint elements a\in A{sa}, every positive map is automatically continuous with respect to the C*-norms and its operator norm equals |\phi(1_A)|_B. A similar statement with approximate units holds for non-unital algebras.
• The set of positive functionals \to\mathbb{C} is the dual cone of the cone of positive elements of A.

Examples

• Every *-homomorphism is completely positive.
• For every linear operator V:H_1\to H_2 between Hilbert spaces, the map L(H_1)\to L(H_2), \ A \mapsto V A V^\ast is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
• Every positive functional \phi:A \to \mathbb{C} (in particular every state) is automatically completely positive.
• Given the algebras C(X) and C(Y) of complex-valued continuous functions on compact Hausdorff spaces X, Y, every positive map C(X)\to C(Y) is completely positive.
• The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let T denote this map on \mathbb{C}^{n \times n}. The following is a positive matrix in \mathbb{C}^{2\times 2} \otimes \mathbb{C}^{2\times 2}: \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 I_2 \otimes T 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. Moreover, the eigenvalues of this matrix are 1,1,1 and −1. (This matrix happens to be the Choi matrix of T, in fact.) Incidentally, a map Φ is said to be co-positive if the composition Φ \circ T is positive. The transposition map itself is a co-positive map.

References

References

1. K. R. Davidson: ''C*-Algebras by Example'', American Mathematical Society (1996), ISBN 0-821-80599-1, Thm. IX.4.1
2. [[Richard Kadison. R.V. Kadison]], [[John Ringrose. J. R. Ringrose]]: ''Fundamentals of the Theory of Operator Algebras II'', Academic Press (1983), ISBN 0-1239-3302-1, Sect. 11.5.21