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Decomposability of extremal positive unital maps on $M_2({\Bbb C})$. (English) Zbl 1116.47033
Bożejko, Marek (ed.) et al., Quantum probability. Papers presented at the 25th QP conference on quantum probability and related topics, Bę dlewo, Poland, June 20--26, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 73, 347-356 (2006).
Let $M_n (\Bbb{C})$ be the algebra of matrices with complex entries. For $m,n$ in $\Bbb{N}$, we have $$M_m (M_n (\Bbb{C}) ) \cong M_m (\Bbb{C}) \otimes M_n (\Bbb{C}) \cong M_{mn} (\Bbb{C}).$$ Thus $M_m (\Bbb{M}_n (\Bbb{C}))$ has the structure of a $C^\ast$-algebra. A matrix $A\in M_m (M_n (\Bbb{C}))$ is positive iff $$\sum_{i,j = 1}^m \overline{\mu_i} \mu_j < \vartheta_{i},\ A_{ij} \vartheta_j \leq 0\text{ for every }\vartheta_1, \dots , \vartheta_m \in \Bbb{C}^n\text{ and }\mu_1, \dots, \mu_m \in \Bbb{C}.$$ A linear map $\varphi: M_m (\Bbb{C}) \rightarrow M_n (\Bbb{C})$ is called positive if $\varphi (A)$ is a positive matrix for every positive matrix $A \in M_m (\Bbb{C})$. If $k \in \Bbb{N}$, $\varphi$ is called $k$-positive map (resp., $k$-copositive map) whenever $[\varphi (A)_{ij})]^k_{i,j = 1}$ (resp., $[\varphi (A)_{j,i})]^k_{i,j =1})$ is a positive element in $M_k (M_n (\Bbb{C}))$ for every positive $[A_{ij}]^k_{i,j =1}$ in $M_k (M_m (\Bbb{C}))$. If $\varphi$ is $k$-positive (resp., $k$-copositive) for every $k \in \Bbb{N}$, then $\varphi$ is called completely positive (resp., completely copositive). A positive map which is a sum of completely positive and completely copositive maps is called decomposable. It is known that if $m = n=2$, then every positive map is decomposable. Given an extremal unital positive map $\varphi : M_2 (\Bbb{C}) \rightarrow M_2 (\Bbb{C})$, the authors construct concrete maps $\varphi_1$ and $\varphi_2$ which give a decomposition of $\varphi$. They also show that in most cases this decomposition is unique. For the entire collection see [Zbl 1101.81002].

47B65Positive and order bounded operators
47L07Convex sets and cones of operators
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