Manuscript VII-J-ST@1 · VII-J-ST.tex · theorem · f1f1d82c

Theorem: Complete destructive-clock instrument

Destructive clocks, programme records, and calibration · lines 1256–1286 · proof included or linked

VII-J-ST@1#thm:clock, VII-J-ST@1#eq:clock

The basis change $∣ τ_{a} ⟩ \mapsto ∣ a ⟩$ , followed by destructive Enter on all clock names, has channel

M_{C} (X) = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes D_{a} X D_{a}^{†} = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes Tr_{C} [(P_{a} \otimes I) X (P_{a} \otimes I)] .

It is CPTP on all operators, including every reference extension. Its output has no quantum $C$ factor, and

Tr_{R} M_{C} (X) = Tr_{C} X .

For branch probability $p_{a} > 0$ , the conditional surviving state is $D_{a} ρ D_{a}^{†} / p_{a}$ . Zero-probability branches are not normalized.

Proof. Kraus operators are $∣ a ⟩_{R} \otimes D_{a}$ , with squared moduli summing to $\sum_{a} P_{a} \otimes I = I$ . Expansion in the clock basis shows that both expressions in (eq:clock) return its $aa$ block. Summation gives the partial trace. The calculation is unchanged after adjoining an arbitrary untouched factor. The destructive output type removes the measured quantum names.

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VII-J-ST@1#eq:clock · line 1260
$M_{C} (X) = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes D_{a} X D_{a}^{†} = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes Tr_{C} [(P_{a} \otimes I) X (P_{a} \otimes I)] .$
unlabeled · line 1269
$Tr_{R} M_{C} (X) = Tr_{C} X .$

Theorem: Complete destructive-clock instrument

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