Proposition: Independent ancillary transfer

Let $\rho ,\sigma$ be ancillary states independent of a message and its reference. For a fixed channel $\Lambda$, define

Then

Here and below $d_{\diamond }=\frac{1}{2}\text{norm}{\cdot }_{\diamond }$.

Proof. For every referenced input state $\xi$, the input difference is $\xi \otimes (\rho -\sigma )$, of trace norm $\text{norm}{\rho -\sigma }_1$. Apply channel contraction and take the supremum.