Theorem: Sharp reference-stable numerical-report error

With $d_{\diamond }(\Phi ,\Psi )=\frac{1}{2}{\Vert \Phi -\Psi \Vert }_{\diamond }$ and one common admissible $B>0$,

These equalities include arbitrary finite untouched references.

Proof. The difference of the two qubit states has eigenvalues $\pm |v_j-{\tilde{v}}_j|/(2B)$, and hence trace norm $|v_j-{\tilde{v}}_j|/B$. For any operator $X$, including a reference extension, the task blocks are orthogonal. Thus the output difference has trace norm

Clock pinching is an average of unitary conjugations, so ${\sum }_j{\Vert X_{jj}\Vert }_1\le {\Vert X\Vert }_1$. This proves the diamond upper bound. A basis clock input at a maximizing index attains it. For binary reports, the two scalar differences are $\pm (r_j-{\tilde{r}}_j)/(2B)$; the same calculation applies.