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?
Best Answer
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.