Lemma: Binary reduction and attainment

The strength is the maximum of the absolute difference of one receiver-event probability. It lies in $[0,1]$ and is attained.

Proof. For finite probability laws, total variation equals the maximum absolute difference over subsets of outcomes. Aggregate such a subset into one binary instrument element.

The Choi operators of channels and binary instrument elements form closed bounded sets in finite-dimensional spaces. For a binary element $B$, the inequalities are

Every such $B$ can be completed to an instrument: if $R=I-{\mathrm{Tr}}_{\mathrm{out}}B\ge 0$, add $R\otimes \tau$, where $\tau$ is a fixed output state. The resulting optimization is continuous on a compact set, proving attainment.