Lemma: Exact distance between binary record channels

Let $E,F$ be effects on the same finite-dimensional Hilbert space. For their binary record channels,

Proof. Set $D=E-F$. With any reference, the output difference has blocks $-Y,Y$, where

for an input trace-class operator $X$. Trace-norm duality gives

The two output blocks have total trace norm $2\Vert Y{\Vert }_1$. This proves the upper bound on the diamond norm. A pure eigenstate of the self-adjoint matrix $D$ corresponding to an eigenvalue of largest absolute value attains that bound without a reference.