On circle preserving quadratic operators
Let \(U : M_2(\mathbb C) \otimes M_2(\mathbb C) \to M_2(\mathbb C) \otimes M_2(\mathbb C)\) be the linear operator such that \(U(x \otimes y) = y \otimes x\) for all \(x, y \in M_2(\mathbb C)\). A linear operator \(\Delta : M_2(\mathbb C) \to M_2(\mathbb C) \otimes M_2(\mathbb C)\) is said to be a quantum quadratic operator (q.q.o.)\ if it is unital and positive, i.e., \(\Delta x \geq 0\) whenever \(x \geq 0\). The operator \(\Delta\) is a quasi-quantum quadratic operator (quasi q.q.o.)\ if is is unital, \(*\)-preserving and \(V_{\Delta}(\varphi)\in M_2(\mathbb C)\), where \(\varphi \in M_2(\mathbb C)_+^*\). Also, it is symmetric if one has \(U \Delta=\Delta\). A quasi q.q.o.\ \(\Delta\) is called \(q\)-pure if for any pure state \(\varphi\), the state \(V _{\Delta}(\varphi)\) is also pure. In the present paper, the author shows that any \(q\)-pure anti-symmetric quasi q.q.o.\ is not positive. He considers quadratic operators, which preserve the unit circle, and shows that the corresponding quasi q.q.o.\ cannot be positive. He also classifies \(q\)-pure circle preserving quadratic operators into three disjoint classes. Moreover, he proves that a quasi q.q.o.\ corresponding to the first class is block positive.
scientific article; zbMATH DE number 6706530
scientific article