Completely positive map
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.
Definition
Let [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] be C*-algebras. A linear map [math]\displaystyle{ \phi: A\to B }[/math] is called positive map if [math]\displaystyle{ \phi }[/math] maps positive elements to positive elements: [math]\displaystyle{ a\geq 0 \implies \phi(a)\geq 0 }[/math].
Any linear map [math]\displaystyle{ \phi:A\to B }[/math] induces another map
- [math]\displaystyle{ \textrm{id} \otimes \phi : \mathbb{C}^{k \times k} \otimes A \to \mathbb{C}^{k \times k} \otimes B }[/math]
in a natural way. If [math]\displaystyle{ \mathbb{C}^{k\times k}\otimes A }[/math] is identified with the C*-algebra [math]\displaystyle{ A^{k\times k} }[/math] of [math]\displaystyle{ k\times k }[/math]-matrices with entries in [math]\displaystyle{ A }[/math], then [math]\displaystyle{ \textrm{id}\otimes\phi }[/math] acts as
- [math]\displaystyle{ \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}. }[/math]
We say that [math]\displaystyle{ \phi }[/math] is k-positive if [math]\displaystyle{ \textrm{id}_{\mathbb{C}^{k\times k}} \otimes \phi }[/math] is a positive map, and [math]\displaystyle{ \phi }[/math] is called completely positive if [math]\displaystyle{ \phi }[/math] is k-positive for all k.
Properties
- Positive maps are monotone, i.e. [math]\displaystyle{ a_1\leq a_2\implies \phi(a_1)\leq\phi(a_2) }[/math] for all self-adjoint elements [math]\displaystyle{ a_1,a_2\in A_{sa} }[/math].
- Since [math]\displaystyle{ -\|a\|_A 1_A \leq a \leq \|a\|_A 1_A }[/math] every positive map is automatically continuous with respect to the C*-norms and its operator norm equals [math]\displaystyle{ \|\phi(1_A)\|_B }[/math]. A similar statement with approximate units holds for non-unital algebras.
- The set of positive functionals [math]\displaystyle{ \to\mathbb{C} }[/math] is the dual cone of the cone of positive elements of [math]\displaystyle{ A }[/math].
Examples
- Every *-homomorphism is completely positive.
- For every linear operator [math]\displaystyle{ V:H_1\to H_2 }[/math] between Hilbert spaces, the map [math]\displaystyle{ L(H_1)\to L(H_2), \ A\mapsto VAV^\ast }[/math] is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
- Every positive functional [math]\displaystyle{ \phi:A\to\mathbb{C} }[/math] (in particular every state) is automatically completely positive.
- Every positive map [math]\displaystyle{ C(X)\to C(Y) }[/math] 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 [math]\displaystyle{ \mathbb{C}^{n\times n} }[/math]. The following is a positive matrix in [math]\displaystyle{ \mathbb{C}^{2\times 2} \otimes \mathbb{C}^{2\times 2} }[/math]:
- [math]\displaystyle{ \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}. }[/math]
The image of this matrix under [math]\displaystyle{ I_2 \otimes T }[/math] is
- [math]\displaystyle{ \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} , }[/math]
- which is clearly not positive, having determinant -1. Moreover, the eigenvalues of this matrix are 1,1,1 and -1.
- Incidentally, a map Φ is said to be co-positive if the composition Φ [math]\displaystyle{ \circ }[/math] T is positive. The transposition map itself is a co-positive map.
See also
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