Sinkhorn–Knopp theorem for rectangular positive maps

Abstract

In this work, we adapt Sinkhorn–Knopp theorem for rectangular positive maps. We extend their concepts of support and total support to these maps. We show that a positive map T:Mk→Mm is equivalent to a doubly stochastic map if and only if T:Mk→Mm is equivalent to a positive map with total support. Moreover, if k and m are coprime then support is sufficient for the equivalence with a doubly stochastic map. This result provides a necessary and sufficient condition for the filter normal form, which is commonly used in Quantum Information Theory. Let A=∑i=1nAi⊗Bi∈Mk⊗Mm be a state and GA:Mk→Mm be the positive map GA(X)=∑i=1nBitr(AiX). We show that A can be put in the filter normal form if and only if GA:Mk→Mm is equivalent to a positive map with total support. We prove that any state A∈Mk⊗Mm≃Mkm such that dim⁡(ker⁡(A))<k−1, if k=m, and dim⁡(ker⁡(A))<min{k,m}, if k≠m, can be put in the filter normal form.