Manuscript Hawk02-B@2 · Hawk02-B.tex · proposition · 95ea744f

Proposition: Time-resolved optimal recovery

Access controls and exact operational effects › Time dephasing preserves the code · lines 1226–1254 · proof included or linked

Hawk02-B@2#prop:time, Hawk02-B@2#eq:timedecoder

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Time dephasing produces

(Δ_{time} N_{N}) (X) = q Tr (X) ∣ vac ⟩ ⟨ vac ∣ + j \sum p_{j} J_{j} X J_{j}^{†} .

Its optimal recovery error remains $q c_{d}$ . A schedule-independent CPTP decoder is

D_{time} (Y) = j \sum J_{j}^{†} Y J_{j} + Tr [(I_{R} - j \sum J_{j} J_{j}^{†}) Y] \frac{I _{C}}{d} .

It requires internal-label coherence within a time sector, not coherence between different time sectors.

Proof. Pinching removes only inter-time cross terms. The orthogonal sector embeddings and complement replacement make (eq:timedecoder) CPTP. Its composition is $p id + q T_{I / d}$ . The Bell lower bound of Theorem thm:B3 still applies.

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Hawk02-B@2#eq:timedecoder · line 1236
$D_{time} (Y) = j \sum J_{j}^{†} Y J_{j} + Tr [(I_{R} - j \sum J_{j} J_{j}^{†}) Y] \frac{I _{C}}{d} .$
unlabeled · line 1229
$(Δ_{time} N_{N}) (X) = q Tr (X) ∣ vac ⟩ ⟨ vac ∣ + j \sum p_{j} J_{j} X J_{j}^{†} .$

Proposition: Time-resolved optimal recovery

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