Proposition: Sharp obstruction after the named binary readout

In the specialization $K=1,d=2$, define the second bulk effect

For the already fixed record channel $\mathcal{C}={\mathcal{R}}_B\circ {\mathcal{N}}_B$, no processing of its record determines $\mathrm{Tr}(J\rho )$ for every bulk state. The optimal worst-case scalar expectation error is exactly $1/2$.

Moreover, every CPTP decoder $\mathcal{D}$ from that record to a bulk qubit satisfies

No optimality claim for this quantum-channel lower bound is made.

Proof. By (eq:diagnostic-effect),

The corresponding target probabilities for $J$ are $1$ and $0$. Lemma lem:indistinguishable gives the lower bound $1/2$. The constant estimator $h=1/2$ attains it because every effect probability lies in $[0,1]$.

For a CPTP decoder, both inputs give the same decoded state $\tau$. Since $d_{\mathrm{tr}}(|0⟩⟨0|,|1⟩⟨1|)=1$, the trace-distance triangle inequality implies

Worst-case input-state distance is a lower bound on half-diamond distance, proving the channel assertion.