Proposition: Complete restoration

For every complete isometry $V:L\to RB$,

is CPTP and left-inverts ${\mathrm{Ad}}_V$, for any fixed diary state ${\omega }_L$. The complete isometry preserves the trace norm of every operator, also with an external reference.

Proof. Both terms are CP. Their traces sum to $\mathrm{Tr}Y$, and substitution gives ${\mathcal{L}}_V(VX{V}^{†})=X$. An isometric embedding preserves all nonzero singular values, including after tensoring with a reference identity.