# Reference on completely positive maps which are isometries

Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a finite-dimensional Hilbert space $K$. Let $\Phi^\dagger:\mathcal{L}(K)\rightarrow \mathcal{L}(H)$ be the adjoint of $\Phi$, obtained by taking the adjoints of all Kraus operators.

When $\Phi^\dagger\Phi = id_{\mathcal{L}(H)}$, i.e. when $\Phi$ is an isometry in the dagger compact category of finite-dimensional Hilbert spaces and completely positive maps, then it can be shown that $\Phi$ is necessarily in the following form $$\Phi = \rho \mapsto \sum_i q_i V_i \rho V_i^\dagger$$ where $q_i \in \mathbb{R}^+$ and $V_i: H \rightarrow K$ are isometries---in the dagger compact category of finite-dimensional Hilbert spaces and complex linear maps---with pairwise orthogonal images, i.e. $V_j^\dagger V_i = \delta_{ij} id_{H}$.

I already know how to prove the result, but I have the distinct feeling that I have seen it somewhere before, either by itself or as a consequence of some more general characterisation of completely positive maps. Hence the question: is anyone aware of some piece of literature where this result is proven, or from which this result follows as a direct consequence?

(I spend hours trying to track it down, to no avail, but in the process I have discovered a question from a few years ago that is related to it, namely When is this map completely positive?)

Namely, they show that every linear isometry between matrix algebras, $\varphi : M_n(\mathbb C) \rightarrow M_m(\mathbb C)$ has the form $$\varphi(a) = UaV \quad \textrm{or} \quad \varphi(a) = Ua^TV, \quad \forall a\in M_n(\mathbb C)$$ where $U$ and $V$ are unitaries in $M_m(\mathbb C)$.
If you require that $\varphi$ is a completely positive as well then there can be no transpose and $V = U^\dagger$.