Manuscript X-F-ST@1 · X-F-ST.tex · theorem · e71f03ef

Theorem: Complete clock instrument

Finite supplied evaluations and the complete Enter interface › Complete destructive clock · lines 2269–2302 · proof included or linked

X-F-ST@1#thm:clock, X-F-ST@1#eq:clock

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The map

M (X) = j \sum ∣ j ⟩ ⟨ j ∣_{R} \otimes D_{j} X D_{j}^{†} = j \sum ∣ j ⟩ ⟨ j ∣_{R} \otimes Tr_{C_{clk}} [(P_{j} \otimes I_{A}) X (P_{j} \otimes I_{A})]

is CPTP, and

Tr_{R} M (X) = Tr_{C_{clk}} X .

These are identities on all operators, including every untouched reference extension. The clock names are consumed; no quantum clock factor is returned.

Proof. The recorded Kraus maps are $∣ j ⟩_{R} \otimes D_{j}$ , whose squared moduli sum to $\sum_{j} P_{j} \otimes I_{A} = I$ . Expand $X = \sum_{j, l} ∣ τ_{j} ⟩ ⟨ τ_{l} ∣ \otimes X_{j l}$ . Both $j$ -th blocks equal $X_{j j}$ ; their sum is the partial trace. The identical calculation includes any reference. Consumption is the declared destructive interface, whose quantum codomain contains $A$ , not the measured clock.

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X-F-ST@1#eq:clock · line 2272
$M (X) = j \sum ∣ j ⟩ ⟨ j ∣_{R} \otimes D_{j} X D_{j}^{†} = j \sum ∣ j ⟩ ⟨ j ∣_{R} \otimes Tr_{C_{clk}} [(P_{j} \otimes I_{A}) X (P_{j} \otimes I_{A})]$
unlabeled · line 2281
$Tr_{R} M (X) = Tr_{C_{clk}} X .$

Theorem: Complete clock instrument

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