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On the description of bistochastic Kadison-Schwarz operators on $\Bbb M_2(\Bbb C)$. (English) Zbl 1207.81009
Let $A,B$ be $C^*$-algebras and let $T:A \to B$ be a linear map. It is said to be positive if $T(a^*a)\geq 0$ for all $a \in A$, Kadison-Schwarz if $T(a^*a)- T(a)^* T(a)\geq 0$ for all $a \in A$, and completely positive if for each $n \in N$ the map $T \otimes I_n:M_n(A) \to M_n(B)$ is positive. It is well-known that completely positive maps (in fact, even 2-positive maps) are automatically Kadison-Schwarz and that Kadison-Schwarz maps are positive, and in general none of the reversed implications hold. In the reviewed paper the authors use elementary techniques to characterise bistochastic (i.e. unital and trace preserving) Kadison-Schwarz maps $T: M_2 \to M_2$ and use this characterisation together with the results of [Linear Algebra Appl. 347, No. 1--3, 159--187 (2002; Zbl 1032.47046)] to provide explicit examples of bistochastic Kadison-Schwarz maps acting on the 2 by 2 matrices which are not completely positive.

81P15Quantum measurement theory
15A04Linear transformations, semilinear transformations (linear algebra)
46L60Applications of selfadjoint operator algebras to physics
81P45Quantum information, communication, networks
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