Manuscript IX-C-ST@1 · IX-C-ST.tex · theorem · 892f50c4

Theorem: Clock instrument

Shared finite clock, certificates, and regular transport › Complete destructive readout · lines 716–742 · proof included or linked

IX-C-ST@1#thm:instrument, IX-C-ST@1#eq:clock

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The channel

M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})]

is CPTP and satisfies

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

All identities hold with every untouched reference. The quantum clock names are consumed.

Proof. The recorded Kraus operators are $∣ k ⟩_{R} \otimes D_{k}$ . Their squared moduli sum to $I_{C S}$ . Expansion in the calibration basis identifies the $k$ -th output block with the $k k$ -input block and proves both formulae and the record-discard identity. Tensoring those identities proves the reference extension. There is no quantum $C$ -output.

Displayed formulas in this passage

IX-C-ST@1#eq:clock · line 719
$M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})]$
unlabeled · line 727
$Tr_{R} M_{C} (X) = Tr_{C} X .$

Theorem: Clock instrument

Displayed formulas in this passage

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