A new bound on quantum Wielandt inequality

Abstract

A new bound on quantum version of Wielandt inequality for positive (not necessarily completely positive) maps has been established. Also bounds for entanglement breaking and PPT channels are put forward which are better bound than the previous bounds known. We prove that a primitive positive map $\mathcal{E}$ acting on $\mathcal{M}_d$ that satisfies the Schwarz inequality becomes strictly positive after at most $2(d-1)^2$ iterations. This is to say, that after $2(d-1)^2$ iterations, such a map sends every positive semidefinite matrix to a positive definite one. This finding does not depend on the number of Kraus operators as the map may not admit any Kraus decomposition. The motivation of this work is to provide an answer to a question raised in the article \cite{Wielandt} by Sanz-Garc\'ia-Wolf and Cirac.