Spectral arc length of a unitary operator

The spectral arc length $\theta (U)$ is the length of the shortest closed unit-circle arc containing every eigenvalue of $U$.

For $U_{\gamma }=\mathrm{diag}(1,e^{i\gamma })$ with $0\le \gamma <2\pi$,

The convex hull of finitely many unit-circle points contains zero exactly when the points do not lie in any open semicircle, equivalently when their shortest containing arc has length at least $\pi$.

Two unitary gates are perfectly distinguishable in one use exactly when