Theorem: Effect reconstruction, including references

For every boundary effect $G$,

Thus exact reconstruction of this binary readout is equivalent to ${\mathcal{N}}^{\ast }(G)=F$.

For a specified quantum decoder, $G={\mathcal{D}}^{\ast }(F)$ is an effect and

For any Hermitian $O\in \mathcal{B}(C)$, with spectral width $w(O)={\lambda }_{\max }(O)-{\lambda }_{\min }(O)$,

Proof. Put $H={\mathcal{N}}^{\ast }(G)-F$. On a referenced state ${\omega }_{QC}$, the measurement-channel difference has two diagonal blocks $K$ and $-K$, where

For Hermitian $L$ with ${\Vert L\Vert }_{\infty }\le 1$,

Trace-norm duality gives ${\Vert K\Vert }_1\le {\Vert H\Vert }_{\infty }$. The output half-trace norm is ${\Vert K\Vert }_1$. An unreferenced eigenstate attaining the largest absolute eigenvalue of $H$ gives equality. The characterization of channel distance by referenced states then proves (eq:effectdiamond).

Postprocessing contraction proves (eq:effectfromcode). For (eq:observablefromcode), subtract the scalar midpoint $c{I}_C$, where $c=({\lambda }_{\max }(O)+{\lambda }_{\min }(O))/2$. Both channel adjoints fix identity. For every input state, trace duality and the output trace-distance bound give expectation error at most

The difference operator is Hermitian, so its operator norm is the supremum of these absolute expectation errors.