Natural Ordering of the Class of Hermitian Preserving Maps

Question

I am trying to understand Man-Duen Choi's remark 3 in his paper Completely Positive Linear Maps on Complex Matrices:

For a linear map $\Phi : \mathcal{M}_{n} \to \mathcal{M}_{m}$, it is obvious that $\Phi$ is hermitian-preserving iff $(\Phi(E_{jk}))_{jk}$ is hermitian. Endowed with the natural ordering induced by $\mathcal{M}_{n}(\mathcal{M}_{m})$, the class of hermitian-preserving maps is a partially ordered vector space over the reals, while the class of completely positive linear maps is just the positive cone.

Notation: $\mathcal{M}_{n}$ denotes the collection of $n \times n$ complex matrices. $E_{jk} \in \mathcal{M}_{n}$ is the $n \times n$ matrix with a $1$ in the $j,k$ component, and $0$ for all other entries. $\mathcal{M}_{n}(\mathcal{M}_{m})=\mathcal{M}_{m} \otimes \mathcal{M}_{n}$ denotes the collection of $n \times n$ block matrices with $m \times m$ matrices as entries. Alternatively, elements of $\mathcal{M}_{n}(\mathcal{M}_{m})$ can be regarded as an $nm \times nm$ matrix with numerical entries.

My difficulty with understanding the remark stems from the statement about being endowed with a natural ordering...

What does it mean for the class of hermitian-preserving maps to be endowed with the natural ordering induced by $\mathcal{M}_{n}(\mathcal{M}_{m})$?

Answer 1

The natural ordering is $\Phi\geq \Psi$ iff $(\Phi(E_{jk}))_{jk}-(\Psi(E_{jk}))_{jk}\in M_{nm}$ is a positive semidefinite Hermitian matrix. Notice that $\Phi\geq 0$ iff $\Phi$ is a completely positive map.