Unitary gate discrimination

To distinguish two gates in one use without an ancilla, choose an input $|\psi ⟩$ and discriminate the output states $U_1|\psi ⟩$ and $U_2|\psi ⟩$.

Perfect discrimination is possible exactly when some unit vector satisfies

The numerical range of an operator $A$ is

For a finite-dimensional normal matrix, the numerical range is the convex hull of its eigenvalues.

If $Uv=\lambda v$ and $U$ is unitary, norm preservation gives $|\lambda |=1$.

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