A note on the quantum Wielandt inequality

Abstract

In this note, we prove that the index of primitivity of any primitive unital Schwarz map is at most $2(D-1)^2$, where $D$ is the dimension of the underlying matrix algebra. This inequality was first proved by Rahaman for Schwarz maps which were both unital and trace preserving. As we show, the assumption of unitality is basically innocuous, but in general not all primitive unital Schwarz maps are trace preserving. Therefore, the precise purpose of this note is to showcase how to apply the method of Rahaman to unital primitive Schwarz maps that don't preserve trace. As a corollary of this theorem, we show that the index of primitivity of any primitive 2-positive map is at most $2(D-1)^2$, so in particular this bound holds for arbitrary primitive completely positive maps. We briefly discuss of how this relates to a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac.