Theorem: Exact joint assignment error and readout budget

For ideal destructive readout followed only by $W_s$,

For the complete pointer model above, its error against the corresponding ideal readout is at most

Proof. The assignment instrument is

An isometry realizing its assignment sends $|\mathbf{j}⟩$ to ${\sum }_y\sqrt{W_s(y\mid \mathbf{j})}|y⟩|\mathbf{j},y{⟩}_D$, with orthonormal environmental labels.

For any input operator with a reference, pinching in the computational basis is trace-norm contractive. Its diagonal blocks $X_{\mathbf{j}\mathbf{j}}$ obey ${\sum }_{\mathbf{j}}\Vert X_{\mathbf{j}\mathbf{j}}{\Vert }_1\le \Vert X{\Vert }_1$. The complete norm of the assignment difference is therefore at most

A basis input attaining the maximum gives equality.

On a ready branch, replacing the independent pointer preparation costs $p_A$. Lemma lem:stability gives ${\delta }_{R,s}$ for the actual joint pointer evolution. Assignment then costs ${\delta }_{\mathrm{as},s}$. Convexity prices the visible failure branch by $r_F$, proving (eq:readbudget).