Example: One thermal output does not determine a channel

Let $R=C$, choose any full-rank Gibbs state $\omega$ on $C$, and compare ${\mathcal{N}}_1={\mathrm{id}}_C$ with ${\mathcal{N}}_2={\mathcal{T}}_{\omega }$. For the prepared input $\omega$,

Nevertheless

Both channels admit complete finite isometric dilations. Thus equality of the entire accessible state for one preparation, not merely equality of a spectrum, fails to determine recovery.

Proof. The recovery values follow from identity recovery and Lemma lem:constant-error. A dilation of ${\mathrm{id}}_C$ is $|\psi ⟩\mapsto |\psi {⟩}_R|0{⟩}_B$. A dilation of ${\mathcal{T}}_{\omega }$ is $|\psi ⟩\mapsto |{\chi }_{\omega }{⟩}_{RP}|\psi {⟩}_L$ with $B=P\otimes L$, where ${\chi }_{\omega }$ purifies $\omega$.