Corollary: Some constructed setting detects every finite departure

Let ${\tau }_{\theta }={\mathcal{B}}_{\mathbf{a}}({{\rho }_{12}}^{(\theta )})$, ${\tau }_0={\mathcal{B}}_{\mathbf{a}}({{\rho }_{12}}^{(0)})$, and $D=d_{\mathrm{tr}}({\tau }_{\theta },{\tau }_0)$. There is a nonidentity Pauli string $P$ for which the effect $(I+P)/2$ has ideal probability gap at least

Its actual gap is at least this value minus its two complete implementation errors.

Proof. Expand the traceless difference in the Pauli basis. Writing $\Delta t_P=\mathrm{tr}(P({\tau }_{\theta }-{\tau }_0))$,

The effect gap is $|\Delta t_P|/2$. The constructed setting measures its sign event, and Theorem thm:discrimination gives the actual bound.