Normal Form of Linear Maps Between Matrices I am looking for a reference which features the following result -- for lack a better term I will call this "normal form" (of a linear map between matrices) -- and which explores its consequenes.

Let $\ell,m,n,p\in\mathbb N$. For every linear map $\Phi:\mathbb K^{\ell\times m}\to\mathbb K^{n\times p}$ -- which is sometimes called "superoperator" -- there exist $X_{ij}\in\mathbb K^{n\times\ell}$, $Y_{ij}\in\mathbb K^{m\times p}$, $i=1,\ldots,p$, $j=1,\ldots,m$ such that
$$
\Phi(A)=\sum_{i=1}^p\sum_{j=1}^mX_{ij}AY_{ij}
$$
for all $A\in\mathbb K^{\ell\times m}$.

This is a straightforward calculation once one specifies $Y_{ij}:=e_je_i^\top$, $X_{ij}:=\sum_{a=1}^n\sum_{b=1}^\ell(e_a^\top\Phi(e_be_j^\top)e_i)e_ae_b^\top$. A direct consequence of this is $(\mathbb K^{n\times\ell})^{pm}\cong\mathcal L(\mathbb K^{\ell\times m},\mathbb K^{n\times p})$ by means of the linear map $(X_{ij})_{i=1,j=1}^{p,m}\mapsto\sum_{i=1}^p\sum_{j=1}^m X_{ij}(\cdot)e_je_i^\top$. In particular, this map is closed.
My interest in this form comes from the fact that properties of $\Phi$ are frequently encoded in the matrices $X_{ij},Y_{ij}$, e.g., in quantum information theory.
As an example: in the special case $\ell=m$, $n=p$, $\mathbb K=\mathbb C$ -- so $\Phi$ is a linear map between complex square matrices -- expressing $\Phi$ via a "normal form" $\sum_j X_j(\cdot)Y_j$ shows that

*

*$\Phi$ is trace-preserving (i.e. $\operatorname{tr}(\Phi(A))=\operatorname{tr}(A)$ for all $A\in\mathbb C^{m\times m}$) if and only if $\sum_jY_jX_j={\bf 1}_m$

*If $\Phi$ is positive (i.e. positive semi-definite matrices are invariant cone), then $\sum_jX_jY_j,\sum_jY_jX_j\geq 0$. Beyond such necessary criteria, I doubt that one can characterize positivity in this framework because this is a notoriously hard problem (but I will let myself be surprised!)

*$\Phi$ is completely positive if and only if $Y_j=X_j^*$ (for some $(X_j)_j$, $(Y_j)_j$ which make up $\Phi$ via the normal form) as a consequence of Choi's theorem. These $X_j$ are also called Kraus operators
The "closest" subject I know of are linear preserver problems as there one asks when a map $A\mapsto MAN$ is a linear preserver of some function (e.g., the determinant) or other object (e.g., a matrix group) of interest. However, my knowledge on this literature is quite limited so if the above "normal form" is something people frequently deal with there I'm not aware of it.
Anyway, all pointers and comments are appreciated!

Edit: I just found this form in the book "The Theory of Quantum Information" (2018) by Watrous, Proposition 2.20 ff.; his proof idea is essentially the same as what I did: first translate $\Phi$ into the superoperator $\hat\Phi$ which represents the action of $\Phi$ on vectorized inputs, and then translate $\hat\Phi$ back which leads to a linear combination of maps $A\mapsto XAY$.
A more general, and much older result -- assuming square matrices -- is Lemma 2.2 in the famous paper by Gorini, Kossakowski, and Sudarshan from 1976 which states that given a linear map $\Phi:\mathbb C^{n\times n}\to\mathbb C^{n\times n}$ and any orthonormal basis ${F_\alpha}$ of $\mathbb C^{n\times n}$ (w.r.t. the Hilbert-Schmidt inner product) there exists a unique matrix $C\in\mathbb C^{n^2\times n^2}$ such that
$$
\Phi(A)=\sum_{\alpha,\beta}c_{\alpha\beta}F_\alpha AF_\beta^*\,.
$$
(Actually, $c_{\alpha\beta}= \sum_k\operatorname{tr}(F_\beta G_k^*F_\alpha\Phi(G_k))$ where $\{G_k\}_k$ is any orthonormal basis of $\mathbb C^{n\times n}$).
Either way, this of course yields the above form when defining $G_\alpha:=\sum_{\beta}c_{\alpha\beta}F_\beta^*$ because then $\Phi(A)=\sum_\alpha F_\alpha AG_\alpha$.
However, I am quite certain that these are not the only appearences of this form in the literature. So if anybody has seen this somewhere else please let me know!
 A: This term "superoperator" feels a bit overly dramatic. Mathematically, $f(X) = ∑_k A_k X B_k^⊤$ simply is a linear map between vector spaces. Now, if you know a bit of linear algebra, then it is clear that any linear map $f：U→V$ can be encoded as element of the tensor product space $V⊗U^*$. We are used to the case $U=ℝ^n$ and $V=ℝ^m$, but there is nothing really different when considering $U=^{l×m}=^l ⊗ ^m$ and $V=^{n×p} = ^n⊗^p$.
So, let $T_f = ∑_{i=1}^r |v_i⟩⟨u_i| ∈ V⊗U^*$ be the tensor representation of $f$. Note that
$$\tag{1} V⊗U^* = (^n⊗^p)⊗(^l ⊗ ^m)^* ≅ \big(^n⊗(^l)^*\big)⊗ \big(^p ⊗ (^m)^*\big) $$
This trivial isomorphism is really all that is happening here.
$$\begin{aligned} 
f(X) 
&= ∑_{i=1}^r A^{(i)}X(B^{(i)})^⊤
\\&= \Big(∑_{i=1}^r∑_{kl} A_{mk}^{(i)}X_{kl}B_{nl}^{(i)}\Big)_{mn} 
\\&= \Big(∑_{kl} \Big(∑_{i=1}^rA_{mk}^{(i)}B_{nl}^{(i)}\Big) X_{kl}\Big)_{mn} 
  = \Big(∑_{kl} \Big(∑_{i=1}^r A^{(i)}⊗B^{(i)}\Big) X_{kl}\Big)_{mn}
\\&= \Big(∑_{kl} (T_{f})_{mn, kl} X_{kl}\Big)_{mn} 
\\&= T_f ⋅ X
\end{aligned}$$
Where $T_f∈V⊗U^*$ is the tensor representation of $f$ and $\text{"⋅"}：(V⊗U^*) × U → V$ is the tensor contraction that sends $\big(|v⟩⟨u|，|u'⟩\big)$ to $⟨u∣u'⟩|v⟩ ∈ V$.

The aforementioned properties (1), (2), (3) follow as follows:

*

*Note that $tr(X) = ⟨ ∣X⟩_{^n⊗^n}$ (which is sometimes called Frobenius inner product). Then

$$ tr(X) = tr(f(X)) ⟺ ⟨ ∣X⟩ = ⟨ ∣T_f⋅X⟩ ⟺ ⟨ ∣X⟩ = ⟨T_f^⊤⋅ ∣X⟩$$
Where we made use of the transpose map $V⊗U^* → U⊗V^*$. In particular, when we use the normal form $T_f = ∑_i A_i⊗B_i$, then $T_f^⊤ = ∑_{i=1}^r A_i^⊤⊗B_i^⊤$. In particular, $T_f^⊤⋅ = ∑_{i=1}^r A_i^⊤⋅⋅B_i = ∑_{i=1}^r A_i^⊤B_i$. And obviously $⟨ ∣X⟩ = ⟨T_f^⊤⋅ ∣X⟩ = ⟨∑_{i=1}^r A_i^⊤B_i∣X⟩$ for all $X$ if and only if $ = ∑_{i=1}^r A_i^⊤B_i$.


*Note that $X$ is positive definite if and only if $⟨uu^⊤∣X⟩_{^n⊗^n}>0$ for all $u≠0$, or equivalently if $⟨S∣X⟩_{^n⊗^n}>0$ for all positive definite $S$. So $f$ is positive if and only if $∀u≠0: ⟨uu^⊤∣X⟩>0 ⟹ ∀v≠0:⟨vv^⊤∣f(X)⟩>0$. Again, we can simply make use of the dual tensor:

$$⟨vv^⊤∣f(X)⟩ =⟨T_f^⊤⋅vv^⊤∣X⟩ =  ⟨∑_i A_i^⊤vv^⊤B_i∣X⟩ =  ⟨∑_i (A_i^⊤v)(B_i^⊤v)^⊤∣X⟩ $$
In particular, we get positivity if $∑_i A_i^⊤vv^⊤B_i$ is symmetric for all $V$ which is the case when $∑_i A_i⊗B_i = ∑_i B_i ⊗ A_i$, i.e. $T_f$ is a symmetric tensor. In particular, in this case $f() = ∑_i A_i B_i^⊤ = ∑_i B_i A_i^⊤$ must be positive definite.


*Complete positivity: tbd.

