Lemma: Faithful-syndrome uniqueness

Let $d\ge 1$. If a CPTP channel $\mathcal{R}:\mathcal{B}({\mathcal{H}}_a\otimes {\mathbb{C}}^d)\to \mathcal{B}({\mathcal{H}}_a)$ satisfies

for every $\rho$, then $\mathcal{R}={\mathrm{Tr}}_{{\mathbb{C}}^d}$.

Proof. Take Kraus operators $K_{\ell }$ of $\mathcal{R}$. The operators

are a Kraus family for ${\mathrm{id}}_a$. The Choi operator of the identity has rank one. Since it is a sum of positive rank-one operators, every $\mathrm{vec}(L_{\ell j})$ is proportional to $\mathrm{vec}(I_a)$. Thus $L_{\ell j}=c_{\ell j}{I}_a$, and

for some vectors $w_{\ell }$. Trace preservation gives ${\sum }_{\ell }|w_{\ell }⟩⟨w_{\ell }|=I_d$. Substitution yields the partial trace on every operator.