Manuscript IX-D-ST@1 · IX-D-ST.tex · theorem · b1dc631a

Theorem: Complete destructive clock

Shared finite Enter certificate interface › Clock and description register · lines 2238–2262 · proof included or linked

IX-D-ST@1#thm:instrument

The map

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) X (P_{k} \otimes I)]

is CPTP and obeys

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

These identities hold after adjoining every untouched reference. The clock has no quantum output factor.

Proof. The recorded Kraus operators $∣ k ⟩_{R} \otimes D_{k}$ have squared moduli summing to $I_{C S}$ . Clock-basis expansion identifies the $k$ -block with $X_{k k}$ , proving the formulae and the discard identity. Tensoring the calculation proves the reference assertion.

Displayed formulas in this passage

unlabeled · line 2241
$M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I) X (P_{k} \otimes I)]$
unlabeled · line 2248
$Tr_{R} M_{C} (X) = Tr_{C} X .$

Theorem: Complete destructive clock

Displayed formulas in this passage

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