Sinkhorn-Knopp theorem for rectangular positive maps

TL;DR

This paper extends the Sinkhorn-Knopp theorem to rectangular positive maps, providing necessary and sufficient conditions for their equivalence to doubly stochastic maps, with applications in quantum information theory and entanglement detection.

Contribution

It generalizes the Sinkhorn-Knopp theorem to rectangular positive maps, establishing conditions for their equivalence to doubly stochastic maps and applying these results to quantum state normalization.

Findings

Characterization of when a positive map is equivalent to a doubly stochastic map.

Conditions for putting quantum states into filter normal form.

Deeper connection between capacities of rectangular and square positive maps.

Abstract

In this work, we adapt Sinkhorn-Knopp theorem for rectangular positive maps $(T:M_k\rightarrow M_m)$. We extend their concepts of support and total support to these maps. We show that a positive map $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $T:M_k\rightarrow M_m$ is equivalent to a positive map with total support. Moreover, if $k$ and $m$ are coprime then $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $T:M_k\rightarrow M_m$ 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 $A=\sum_{i=1}^nA_i\otimes B_i\in M_k\otimes M_m$ be a state and $G_A: M_k\rightarrow M_m$ be the positive map $G_A(X)=\sum_{i=1}^nB_itr(A_iX)$. We show that $A$ can be put in the filter normal form if and only if $G_A: M_k\rightarrow M_m$ is equivalent to a positive map with total support. We prove that any state $A\in M_k\otimes M_m\simeq M_{km}$ such that $\dim(\ker(A))<k-1$, 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 $T:M_k\rightarrow M_m$ and the capacity of a certain square positive map $\widetilde{T}:M_{mk}\rightarrow M_{mk}$ was noticed. Here, we obtain a deeper connection between these maps. As a corollary of our main results, we prove that $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $\widetilde{T}:M_{mk}\rightarrow M_{mk}$ is equivalent to a doubly stochastic map.