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Fixed points of dual quantum operations. (English) Zbl 1230.81006
Let $$B(H)$$ and $$K(H)$$ be the set of all bounded linear operators and the set of all compact operators on a separable Hilbert space $$H$$, respectively. K. Kraus [Ann. Physics 64, 311–335 (1971; Zbl 1229.81137)] showed that any contractive, normal, and completely positive map $$\phi$$ on $$B(H)$$ can be expressed as $$\phi (X)=\sum_{i=1}^{\infty} A_i X A_i^*$$, where $$\{A_i\}$$ is a sequence in $$B(H)$$ with $$\sum_{i=1}^{\infty} A_i A_i^* \leq I$$.
For a sequence of operators $$\mathcal{A} = \{A_i \}$$ on $$B(H)$$ satisfying $$\sum_{i=1}^{\infty} A_i A_i^* \leq I$$ and $$\sum_{i=1}^{\infty} A_i^* A_i \leq I$$, one can consider maps $$\phi_\mathcal{A}$$ and $$\phi_\mathcal{A}^*$$ on $$B(H)$$ defined by $$\phi_{\mathcal{A}}(X)=\sum_{i=1}^{\infty}A_i X A_i^*$$ and $$\phi_{\mathcal{A}}^* (X)=\sum_{i=1}^{\infty}A_i^* X A_i$$.
This paper deals with the relation between the fixed points of $$\phi_A$$ and $$\phi_A^*$$. Particularly, the author shows that $$\{B\in K(H): \phi_\mathcal{A}(B)=B\}=\{B \in K(H): \phi_{\mathcal{A}}^* (B)=B \}$$.

##### MSC:
 81P15 Quantum measurement theory, state operations, state preparations 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
##### Keywords:
quantum effect; quantum operation; fixed point
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##### References:
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