Manuscript IV-B@1 · IV-B-v1.tex · theorem · dfb66408

Theorem: Complete clock extraction

The finite destructive clock instrument › Complete identity, references, and consumption · lines 1437–1464 · proof included or linked

IV-B@1#thm:instrument, IV-B@1#eq:drop-record

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The channel (eq:enter) is CPTP. With $D_{k} = ⟨ τ_{k} ∣ \otimes I_{X}$ , it equals

M_{C} (Z) = k \sum ∣ k ⟩ ⟨ k ∣ \otimes D_{k} Z D_{k}^{*} .

For every untouched reference $E$ , this equality remains valid after tensoring with $id_{E}$ . Moreover,

Tr_{R} [(M_{C} \otimes id_{E}) (Z)] = Tr_{C} Z .

The conditional residual on a positive-probability branch is $D_{k} ρ D_{k}^{*} / Tr (D_{k} ρ D_{k}^{*})$ .

Proof. The operators $A_{k} = ∣ k ⟩_{R} \otimes D_{k}$ satisfy

k \sum A_{k}^{*} A_{k} = k \sum P_{k} \otimes I_{X} = I .

They therefore define a CPTP map. Expanding $Z = \sum_{j, ℓ} ∣ τ_{j} ⟩ ⟨ τ_{ℓ} ∣ \otimes Z_{j ℓ}$ shows that both branch formulas equal $Z_{k k}$ . Summing gives (eq:drop-record). The calculation is unchanged with $X$ replaced by $X \otimes E$ .

Displayed formulas in this passage

IV-B@1#eq:drop-record · line 1446
$Tr_{R} [(M_{C} \otimes id_{E}) (Z)] = Tr_{C} Z .$
unlabeled · line 1441
$M_{C} (Z) = k \sum ∣ k ⟩ ⟨ k ∣ \otimes D_{k} Z D_{k}^{*} .$
unlabeled · line 1456
$k \sum A_{k}^{*} A_{k} = k \sum P_{k} \otimes I_{X} = I .$

Theorem: Complete clock extraction

Displayed formulas in this passage

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