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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
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