Theorem: Exact access-dependent optima

For the transfer completion:

where $F_{\mathrm{num}}$ is the fixed destructive number record in the occupation basis of (eq:thermalpair), obtained before $J$. Without $E$, that number record has score $1/2$.

Over all deterministic CPTP decoders from $ER$ to a message qubit,

where $F_e$ is the overlap with a maximally entangled reference–message state. For ${\mathcal{N}}_0$, even with $E$, the optimal delayed-query score is $1/2$, the optimal entanglement fidelity is $1/4$, and the optimal half-diamond recovery error is $3/4$.

Proof. The late marginal is constant by (eq:completionmarginals), proving (eq:lateoptimum). With $E$, the key and pair-sector projectors are mutually orthogonal and can be read without measuring the logical qubit. On each paired sector the decoder applies ${J_{n,k}}^{†}$ and ${P_r}^{†}$; the recovered logical state is ${{\rho }_x}^A$. Unpaired sectors are constant. Equation (eq:flagaverage) therefore gives (eq:jointoptimum), with attainability by query-dependent $X$ or $Z$ measurement on the recovered state.

For a paired sector, destructive number measurement is logical $Z$ measurement, with its outcome flip determined by $r_1$. After the key correction, its distribution has no dependence on $x_1$, so the first requested bit is guessed with probability $1/2$. For $x_2$, the optimal success probability is

Averaging over $J$ gives $1/2+1/(4\sqrt{2})$ on every paired branch. Unpaired branches score $1/2$, proving (eq:numberoptimum). Twirling proves the unassisted number statement.

For referenced recovery, paired sectors permit perfect recovery and contribute Bell overlap at most one. Unpaired sectors are message-independent. After any local decoder their reference–output state has the form $I_{\mathrm{ref}}/2\otimes {\sigma }_{\mathrm{out}}$, whose overlap with a normalized Bell state is $1/4$. Hence every decoder satisfies

The coherent branchwise decoder, preparing $I_A/2$ on an unpaired sector, attains this bound. Its complete decoded channel is

For any decoder, testing the Bell projector gives $D_{\diamond }(\mathcal{D}{\mathcal{N}}_1,{\mathrm{id}}_A)\ge 1-F_e$, establishing the lower bound in (eq:optimaldiamond). The Pauli identity

shows $D_{\diamond }(\mathcal{T},\mathrm{id})\le 3/4$ by the triangle inequality, since each unitary-channel distance is at most one in half-diamond normalization. A Bell input gives four orthogonal Bell outputs and proves the matching lower bound $3/4$. Homogeneity now gives

which is attained by the stated decoder.

The constant channel ${\mathcal{N}}_0$ has no message-dependent output. The same Bell argument gives fidelity $1/4$ and error at least $3/4$; preparing $I_A/2$ attains it.