Theorem: Separate profile channel and observable estimates

On every fixed finite outgoing packet family,

For the fixed target sine effect,

Proof. The conditional vector states in the quasifree realization have overlap modulus $e^{-r^2/2}$; their pure-state trace distance is $\sqrt{1-e^{-r^2}}$. Restriction to packets contracts it. A measure-and-prepare channel with the same source projectors has half-diamond difference at most the maximal conditional state distance: pinching a referenced input into source blocks and applying the triangle inequality proves this bound. Thus it remains valid with references.

For the observable estimate, use the exact sine formula, $|\sin x-\sin y|\le |x-y|$, and $|\sigma (\hat{a}-a,l)|\le 2\Vert \hat{a}-a{\Vert }_0\Vert l{\Vert }_0$.