Lemma: Distance conversions and stability

Let $\Phi ,\Psi$ have input dimension $d$, and define

Then

For common-output isometries,

The same allowance holds after a common partial trace.

CPTP pre- and postprocessing contract these distances. On fixed output spaces,

and

Proof. A normalized pure referenced input can, after an isometry on its reference support, be written as $(L\otimes I)|{\Phi }_d⟩$, where $\mathrm{Tr}(L^{†}L)=d$ and ${\Vert L\Vert }^2\le d$. Multiplication of the normalized Choi difference by $L$ and $L^{†}$ bounds its trace norm by $d$ times the Choi trace norm. Convexity handles mixed inputs. This proves (eq:choibound).

For normalized vectors $u,v$,

Apply this to referenced isometric outputs and then use convexity and partial-trace contraction.

The contraction statements follow from the definition of the diamond norm. For each fixed decoder, the triangle inequality changes its error by at most the channel comparison allowance. Taking infima in both directions proves recovery stability. Distance to the replacement-channel set is likewise one-Lipschitz.