Proposition: Bounded observable transfer

Let $\Phi ,\Psi$ be channels on common complete interfaces with

For a fixed input state and a Hermitian output observable $O$,

The same bound holds when the input includes an untouched reference.

Proof. Let $\Delta$ be the output difference. Then $\mathrm{Tr}\Delta =0$ and $\Vert \Delta {\Vert }_1\le 2ϵ$. Subtract the scalar midpoint $c=({\lambda }_{\max }(O)+{\lambda }_{\min }(O))/2$:

The spectral radius about this midpoint is half the displayed spectral diameter. The diamond norm includes referenced inputs.