Consider the complex $n$-by-$n$ matrices $M\_n$. 
Suppose that $A\_i$, for $i=1,\ldots,n^2$, satisfy $\mathrm{Tr}(A\_i^\*
A\_j)=\delta\_{ij}$, so that together they form an orthonormal basis for
$M\_n$. Define a linear map $T \colon M\_n \to M\_n \otimes M\_n$ by $T(A\_i) = A\_i \otimes A\_i$.



> 
> Question: when is $T$ completely positive?
> 
> 
> 


For example, if $A\_i$ are the matrices with a single entry one and the rest zeroes in some fixed basis of $\mathbb{C}^n$, then $T$ is completely positive. In fact, I think these might be the only examples. If $T$ is completely positive, then the following are equivalent to $A\_i$ being matrix units as in the above example:


* each $A\_i$ has rank one;
* each positive semidefinite $A\_i$ has trace one;
* the set $\{0,A\_1,\ldots,A\_{n^2}\}$ is closed under multiplication;
* $T(1)$ is idempotent;
* $T^\*(1) \leq 1$;
* $T$ preserves trace.


These are sufficient conditions, but proving they are sufficient doesn't use $\mathrm{Tr}(A\_i^\* A\_j)=\delta\_{ij}$ at all. Are they necessary?



When is this map completely positive?
Further to the linear map $T: M\_n \rightarrow M\_n \otimes M\_n$ defined above by setting $T(A\_i):=A\_i \otimes A\_i$, consider the linear map $E: M\_n \rightarrow \mathbb{C}$ defined by setting $E(A\_i) := 1$ for all $i=1,...,n^2$.
Let $\eta: \mathbb{C} \rightarrow M\_n \otimes M\_n^\ast$ and $\epsilon: M\_n^\ast \otimes M\_n \rightarrow \mathbb{C}$ be the cups and caps of CP maps:


$$
\eta(1) := \sum\_{i} A\_i \otimes A\_i^\ast
\hspace{1cm}
\epsilon(A\_i^\ast \otimes A\_j) := \delta\_{ij} 
$$


Also, let $\sigma := M\_n \otimes M\_n^\ast \rightarrow M\_n^\ast \otimes M\_n$ be the swap of CP maps:


$$
\sigma(A\_i \otimes A\_j^\ast) := A\_j^\ast \otimes A\_i
$$


If $T$ is a CP map, then so is $E$, because the latter can be obtained from the former by composition and tensor product with the CP maps $id\_{M\_n}$, $\eta$, $\epsilon$ and $\sigma$:


$$
E = \left(id\_{M\_n}\otimes (\epsilon \circ \sigma)\right) \circ (T \otimes id\_{M\_n}) \circ \eta
$$


By the way $T$ and $E$ are defined, $(T, E, T^\dagger, E^\dagger)$ is a special commutative $\dagger$-Frobenius algebra of linear maps. If $T$ is CP, then $(T, E, T^\dagger, E^\dagger)$ is a special commutative $\dagger$-Frobenius algebra of CP maps.
By Corollary 7 of <https://arxiv.org/abs/2110.07074v2>, such an algebra necessarily arises by "doubling" of a special commutative $\dagger$-Frobenius algebra $(t, e, t^\dagger, e^\dagger)$ of linear maps. The latter are defined as follows, for some choice of OBN $|\phi\_i\rangle$ on $\mathbb{C}^n$:


$$
t(|\phi\_i\rangle) := |\phi\_i\rangle\otimes|\phi\_i\rangle
\hspace{1cm}
e(|\phi\_i\rangle) := 1
$$


Hence, $T$ takes the form $T(|\phi\_i\rangle\langle\phi\_j|) := |\phi\_i\rangle\langle\phi\_j| \otimes |\phi\_i\rangle\langle\phi\_j|$, as originally conjectured.


