Sinkhorn-Knopp theorem for rectangular positive maps

In this work, we adapt Sinkhorn-Knopp theorem for rectangular positive maps $(T:M_k\to M_m)$. We extend their concepts of support and total support to these maps. We show that a positive map $T:M_k\to M_m$ is equivalent to a doubly stochastic map if and only if $T:M_k\to M_m$ is equivalent to a positive map with total support. Moreover, if $k$ and $m$ are coprime then $T:M_k\to M_m$ is equivalent to a doubly stochastic map if and only if $T:M_k\to M_m$ has support. This result provides a necessary and sufficient condition for the filter normal form, which is commonly used in Quantum Information Theory in order to simplify the task of detecting entanglement. Let $A=\sum_{i=1}^nA_i\otimes B_i\in M_k\otimes M_m$beastateand$G_A: M_k\rightarrow M_m$ be the positive map $G_A(X)={\sum }_{i=1}^n{B}_{i}tr(A_{i}X)$. We show that $A$ can be put in the filter normal form if and only if $G_A:M_k\to M_m$ is equivalent to a positive map with total support. We prove that any state $A\in M_k\otimes M_m\simeq M_{km}$suchthat$\dim(\ker(A))<k-1$,if$k=m$,and$\dim(\ker(A))<\min{k,m}$,if$k\neq m$, can be put in the filter normal form. Recently, a connection between the capacity of a rectangular positive map $T:M_k\to M_m$ and the capacity of a certain square positive map $\tilde{T}:M_{mk}\to M_{mk}$ was noticed. Here, we obtain a deeper connection between these maps. As a corollary of our main results, we prove that $T:M_k\to M_m$ is equivalent to a doubly stochastic map if and only if $\tilde{T}:M_{mk}\to M_{mk}$ is equivalent to a doubly stochastic map.