Theorem: Independent ancillary interface

Let $\rho ,\sigma$ be ancillary states in a common space, independent of the message and its reference. For a fixed CPTP map $\Lambda :M\otimes A\to B$, define ${\mathcal{N}}_{\rho }(X)=\Lambda (X\otimes \rho )$ and similarly ${\mathcal{N}}_{\sigma }$. Then

The bound also holds after any common deterministic decoder.

Proof. For a referenced message state ${\xi }_{MR}$, the input difference is ${\xi }_{MR}\otimes (\rho -\sigma )$, with trace norm ${\Vert \rho -\sigma \Vert }_1$. Apply CPTP contraction and take the supremum over references.