Manuscript VII-F-ST@1 · VII-F-ST.tex · theorem · 9eee7a44

Theorem: Complete destructive-clock identity

Finite destructive clock and an exact evaluation table › The complete instrument · lines 1199–1234 · proof included or linked

VII-F-ST@1#thm:instrument, VII-F-ST@1#eq:instrument, VII-F-ST@1#eq:discard

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The channel implemented by a calibrated basis change followed by destructive Enter on all $m$ clock names is

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 completely positive and trace preserving on all operators, including after tensoring with $id_{E}$ . It returns the code and the unnormalized conditional payload, and

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

The clock has no quantum output factor.

Proof. The Kraus operators are $∣ a ⟩_{R} \otimes D_{a}$ , and

a \sum D_{a}^{†} D_{a} = a \sum P_{a} \otimes I_{S} = I_{C S} .

Expand $X = \sum_{a, b} ∣ τ_{a} ⟩ ⟨ τ_{b} ∣ \otimes X_{ab}$ . Both expressions in (eq:instrument) have block $X_{aa}$ . Summing the blocks proves (eq:discard). The calculation includes a reference without change. Consumption is the declared destructive interface: after the measurement the $C$ -names are unavailable as quantum inputs.

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VII-F-ST@1#eq:instrument · line 1203
$M_{C} (X) = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes D_{a} X D_{a}^{†} = a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes Tr_{C} [(P_{a} \otimes I) X (P_{a} \otimes I)] .$
VII-F-ST@1#eq:discard · line 1215
$Tr_{R} M_{C} (X) = Tr_{C} X .$
unlabeled · line 1224
$a \sum D_{a}^{†} D_{a} = a \sum P_{a} \otimes I_{S} = I_{C S} .$

Theorem: Complete destructive-clock identity

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