Manuscript IX-A-ST@1 · IX-A-ST.tex · theorem · 6e99d1b2

Theorem: Destructive clock

Finite evaluation tables and one destructive clock › Complete clock instrument · lines 1445–1471 · proof included or linked

IX-A-ST@1#thm:instrument, IX-A-ST@1#eq:instrument

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Calibrated basis change followed by destructive Enter 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_{S}) X (P_{a} \otimes I_{S})] .

It is CPTP, including with $id_{E}$ , and

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

Its quantum output contains $S$ , not $C$ . On a branch of positive probability one may normalize the returned block; zero branches require no conditional state.

Proof. The Kraus operators are $∣ a ⟩_{R} \otimes D_{a}$ . Their squared moduli sum to $\sum_{a} P_{a} \otimes I_{S} = I$ . Expansion of $X$ in clock matrix units shows that both expressions return its diagonal $aa$ -block. Summing the blocks gives the partial-trace identity. The calculation is unchanged with an additional reference.

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IX-A-ST@1#eq:instrument · line 1448
$M_{C} (X) = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes D_{a} X D_{a}^{†} = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes Tr_{C} [(P_{a} \otimes I_{S}) X (P_{a} \otimes I_{S})] .$
unlabeled · line 1456
$Tr_{R} M_{C} (X) = Tr_{C} X .$

Theorem: Destructive clock

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