Manuscript VIII-ST@1 · VIII-ST.tex · theorem · fe979f3b

Theorem: Complete destructive-clock instrument

One destructive clock and complete table instruments · lines 1835–1863 · proof included or linked

VIII-ST@1#thm:instrument, VIII-ST@1#eq:clock, VIII-ST@1#eq:clock-discard

A calibrated basis change 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 and with every untouched reference. It returns the classical code and the unnormalized conditional surviving block, has no quantum clock output, and obeys

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

Positive branches may be normalized; zero branches are not.

Proof. The Kraus operators are $∣ a ⟩_{R} \otimes D_{a}$ , and their squared moduli sum to $\sum_{a} P_{a} \otimes I = I$ . Expansion in the clock basis shows that each formula in (eq:clock) returns the same $aa$ -block. Summing those blocks proves (eq:clock-discard). The calculation is unchanged after adjoining a reference. Consumption is the declared destructive interface.

Displayed formulas in this passage

VIII-ST@1#eq:clock · line 1839
$M_{C} (X) = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes D_{a} X D_{a}^{†} = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes Tr_{C} [(P_{a} \otimes I) X (P_{a} \otimes I)] .$
VIII-ST@1#eq:clock-discard · line 1848
$Tr_{R} M_{C} (X) = Tr_{C} X .$

Theorem: Complete destructive-clock instrument

Displayed formulas in this passage

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