A new bound on quantum Wielandt inequality

TL;DR

This paper establishes a new upper bound on the quantum Wielandt inequality for positive maps, improving bounds for entanglement breaking and PPT channels, and shows that primitive positive maps become strictly positive after a finite number of iterations.

Contribution

It introduces a novel bound on quantum Wielandt inequality for positive maps that does not depend on Kraus decompositions, addressing a question from prior research.

Findings

Primitive positive maps become strictly positive after at most 2(d-1)^2 iterations.

New bounds for entanglement breaking and PPT channels are tighter than previous bounds.

The bound applies without requiring the map to be completely positive or to have Kraus operators.

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.