Theorem: Positive TV gap on a finite dictionary

A controlled use of the signal instrument gives a complete instrument on every $JQ{A}_{\mathrm{fin}}$ operator. For the preparations (eq:dictionaryinput),

The corresponding complete output trace distance, including $A_{\mathrm{fin}}E$, is the same.

The gap is positive exactly when at least one retained task has ${\pi }_j>0$, $r_j<1$, and ${\nu }_j{d}_j\ne 0$.

Proof. Destructively read $J$, then apply the instrument selected by its classical code. The Kraus maps are tensor products of $⟨j{|}_J$ with the branch maps in Theorem thm:signal. Completeness follows by summing first the branch maps and then the orthogonal task projectors.

For the declared inputs, different $j$-blocks are orthogonal. Their trace norms add, and Theorem thm:TV applies in each block.