Sinkhorn–Knopp theorem for rectangular positive maps
Daniel Cariello · Linear and Multilinear Algebra · 2018
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))