Theorem: Worst-path adaptive bound

Consider ideal and implemented versions of the same finite public-outcome tree, with matched node and terminal interfaces. At each nonterminal node $v$, suppose its complete recorded node channels obey

Include terminal comparison maps as final nodes when needed. Then

An input preparation allowance is added once, before the tree. The bound does not require independent outcome laws.

Proof. At a leaf let the remaining error bound be zero. At an internal node define recursively

Let $C_{va},{\tilde{C}}_{va}$ be the complete continuations at its children. On the classical–quantum direct-sum output, the controlled continuation difference has half-diamond norm at most ${\max }_a{b}_{va}$.

Indeed, after adjoining a reference, a block-diagonal operator $Z={⨁}_a{Z}_a$ has ${\Vert Z\Vert }_1={\sum }_a{\Vert Z_a\Vert }_1$, and

The canonical pinching extension is trace-norm contractive, so the same bound applies on its full matrix presentation.

Insert the ideal node channel between the two complete node-plus-continuation compositions. Contraction gives the node error $e_v$, and the controlled-continuation estimate gives ${\max }_a{b}_{va}$. Backward induction proves the recursion. Its root value is the maximum root-to-leaf sum. Cap at one.