A note on the \(p \rightarrow q\) norms of 2-positive maps
This paper is concerned with a mathematical question raised by \textit{C.\,King} and \textit{M.\,B.\thinspace Ruskai} [Quantum Inf.\ Comput.\ 4, No.\,6--7, 500--512 (2004; Zbl 1162.47307)], namely: Is the \(p \to q\) norm of a completely positive map \(\Phi\) acting between Schatten \(p\) and \(q\) classes of Hermitian operators, \(\|\Phi\|_{p \to q}=\sup_{A=A^*}\frac{\|\Phi (A)\|_q}{\|A\|_p}\), equal to the \(p \to q\) norm of the map when acting between Schatten classes of general, not necessarily Hermitian, operators? A general proof has been given by \textit{J.\,Watrous} [Quantum Inf.\ Comput.\ 5, No.\,1, 57--67 (2005; Zbl 1162.47308)] for any completely positive map and for all values \(p,q\geq 1\). In this paper, the author presents an alternative proof, which shows that the statement holds more generally for all 2-positive maps.
scientific article; zbMATH DE number 5505640
scientific article