Manuscript VII-E-ST@1 · VII-E-ST.tex · theorem · d28d9c5d

Theorem: Complete destructive clock

Finite clock, residual audit, and encoding limits · lines 1927–1964 · proof included or linked

VII-E-ST@1#thm:instrument, VII-E-ST@1#eq:clock-instrument, VII-E-ST@1#eq:clock-audit

With $D_{s} = ⟨ τ_{s} ∣_{C} \otimes I_{S}$ ,

M_{C} (X) = s \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes D_{s} X D_{s}^{†} = s \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes Tr_{C} [(P_{s} \otimes I) X (P_{s} \otimes I)], P_{s} = ∣ τ_{s} ⟩ ⟨ τ_{s} ∣,

is a channel. It returns the classical code and the subnormalized conditional payload, including every untouched reference. It consumes the quantum clock names and obeys

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

If an outcome-selected residual instrument has operators $A_{s, y}$ with $\sum_{y} A_{s, y}^{†} A_{s, y} = I$ , the complete clock–audit map is

X ⟼ s, y \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes A_{s, y} D_{s} X D_{s}^{†} A_{s, y}^{†} .

Proof. The clock Kraus operators are $∣ s ⟩_{R} \otimes D_{s}$ . Their squared moduli sum to $I_{C S}$ . Expansion in the calibration basis proves the partial-trace formula. Summing the diagonal clock blocks proves the record-discard identity. The audit Kraus family is also complete by its stated normalization. All equalities hold on matrix units and remain valid under reference extension.

Displayed formulas in this passage

VII-E-ST@1#eq:clock-instrument · line 1930
$M_{C} (X) = s \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes D_{s} X D_{s}^{†} = s \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes Tr_{C} [(P_{s} \otimes I) X (P_{s} \otimes I)], P_{s} = ∣ τ_{s} ⟩ ⟨ τ_{s} ∣,$
VII-E-ST@1#eq:clock-audit · line 1948
$X ⟼ s, y \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes A_{s, y} D_{s} X D_{s}^{†} A_{s, y}^{†} .$
unlabeled · line 1941
$Tr_{R} M_{C} (X) = Tr_{C} X .$

Theorem: Complete destructive clock

Displayed formulas in this passage

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