Proposition: Benchmark identities

The table is exact.

Proof. For Kraus operators $K_a$,

This gives the overlaps.

For Pauli channels with probabilities $a_P,b_P$, the triangle inequality gives $d_{\diamond }(\Phi ,\Psi )\le \frac{1}{2}{\sum }_P|a_P-b_P|$. Their Bell outputs are mixtures of orthogonal Bell projectors, giving equality. This proves the dephasing and completed-loss errors; the replacement errors were proved above.

For a unitary rotation, a pure referenced input has output distance

The convex hull of the eigenvalues has minimum modulus $\left|\cos (\theta /2)\right|$, attained by equal weights. Convexity handles mixed inputs. Appendix app:AD proves the amplitude-damping distance.