New multiplicativity results for qubit maps

Abstract

Let $Φ$ be a trace-preserving, positivity-preserving (but not necessarily completely positive) linear map on the algebra of complex $2 \times 2$ matrices, and let $Ω$ be any finite-dimensional completely positive map. For $p=2$ and $p \geq 4$, we prove that the maximal $p$-norm of the product map $Φ\ot Ω$ is the product of the maximal $p$-norms of $Φ$ and $Ω$. Restricting $Φ$ to the class of completely positive maps, this settles the multiplicativity question for all qubit channels in the range of values $p \geq 4$.