Lemma: Channel stability and local dictionaries

On common output spaces,

Separate isometric embeddings of $R$ and $B$ preserve both diagnostics.

For common-output isometries,

For channels with input dimension $d$,

Proof. For each decoder, the triangle inequality and CPTP contraction change its error by at most the channel comparison distance. Take infima in both directions. Distance to the replacement set is likewise one-Lipschitz.

An isometry $J:H\to H^{\prime }$ has the CPTP retraction

Embedding candidate decoders or replacement states and then retracting proves invariance.

For referenced input $\rho$, expand the complete-output difference as

with reference identities understood. Its trace norm is at most $2{\Vert V_1-V_2\Vert }_{\infty }$. Partial trace is contractive.

Every pure input with a $d$-dimensional reference is $(L\otimes I)|{\Phi }_d⟩$, with $\mathrm{Tr}(L^{†}L)=d$ and ${\Vert L\Vert }_{\infty }^2\le d$. Its output difference is the Choi difference multiplied on both sides by $L,L^{†}$. Convexity completes the bound.