Lemma: Isometry and recovery stability

Let $V,W:C\to R\otimes B$ be isometries and suppose $\Vert V-W\Vert \le \kappa$. Then

and both corresponding marginal channel distances are at most $\kappa$. Moreover,

and

Proof. For every referenced operator $X$,

with reference identities understood. Its trace norm is at most $2\kappa {\Vert X\Vert }_1$. Taking the stabilized supremum and dividing by two proves the complete-channel estimate. Partial trace gives the marginal estimates.

For every $\mathcal{D}$, the triangle inequality and contraction give

Take infima and interchange $V,W$. The residual estimate is the same argument with each replacement state $\sigma$.