A Note on the p->q norms of Completely Positive Maps

TL;DR

This paper provides an alternative proof that the p→q norm of a completely positive map between Schatten classes is the same whether acting on self-adjoint or general operators, addressing a question posed by King and Ruskai.

Contribution

It offers a new proof confirming the equality of p→q norms for completely positive maps on Schatten classes, complementing Watrous's previous proof.

Findings

Confirmed the equality of p→q norms for self-adjoint and general operators

Provided an alternative proof method for the norm equality

Addressed a question posed by King and Ruskai

Abstract

King and Ruskai asked whether the $p\to q$ norm of a completely positive map $\Phi$, acting between Schatten $p$ and $q$ classes of self-adjoint operators, $||\Phi||_{p\to q} = \max_{A=A^*} \frac{||\Phi(A)||_q}{||A||_p}$, is equal to the $p\to q$ norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.