Proposition: A coherent logical rotation

For $|\theta |\le \pi$ and $U_{\theta }=\exp (-i\theta Z/2)$,

This is the error of the specified uncorrected ideal decoder, not an optimal-decoder claim when $\theta$ is known.

Proof. For a pure input with any reference, the overlap with its rotated version is $\mathrm{Tr}({\rho }_L{U}_{\theta })$. The numerical range of $U_{\theta }$ is the line segment joining its two eigenvalues, whose minimum modulus is $\cos (\theta /2)$. The pure-state trace-distance formula gives the upper bound. The input $|+⟩$ attains it. Convexity handles mixed inputs.