Sinkhorn-Knopp theorem for positive maps

Daniel Cariello

Published 2016-09-22Version 1

A positive map $S:M_k\rightarrow M_m$ is called doubly stochastic if $S(\frac{Id}{\sqrt{k}})=\frac{Id}{\sqrt{m}}$ and $S^*(\frac{Id}{\sqrt{m}})=\frac{Id}{\sqrt{k}}$. Here, we show that, for a positive map $T:M_k\rightarrow M_m$, there are invertible matrices $X'\in M_k$, $Y'\in M_m$ such that $Y'T(X'XX'^*)Y'^*$ is doubly stochastic if and only if $T(Id)$ and $T^*(Id)$ are invertible matrices and there are orthogonal projections $V_i\in M_k$, $W_i\in M_m$, $1\leq i\leq s$, such that $\mathbb{C}^k=\bigoplus_{i=1}^s\Im(V_i)$, $\mathbb{C}^m=\bigoplus_{i=1}^s\Im(W_i)$ and, for every $i$, $T(V_iM_kV_i)\subset W_iM_mW_i$, $\frac{\text{rank}(W_i)}{\text{rank}(V_i)}=\frac{m}{k}$ and $\text{rank}(X)\text{rank}(W_i)<\text{rank}(T(X))\text{rank}(V_i)$ for every positive semidefinite Hermitian matrix $X\in V_iM_kV_i$ with $0<\text{rank}(X)<\text{rank}(V_i)$. In order to obtain this theorem, we generalize Sinkhorn and Knopp ideas of support and total support for positive maps and we adapt their iterative algorithm. This result provides a necessary and sufficent condition for the existence of the filter normal form, which is commonly used in Quantum Information Theory. We prove the existence of this normal form for states $A\in M_k\otimes M_m\simeq M_{km}$ such that $\dim(\ker(A))<k-1$, if $k=m$, and $\dim(\ker(A))<\min\{k,m\}$, if $k\neq m$.