Manuscript X-G-ST@1 · X-G-ST.tex · theorem · 3c97567f

Theorem: Complete clock instrument

Stage 5: supplied evaluations and a complete Enter instrument › Complete destructive clock · lines 2213–2246 · proof included or linked

X-G-ST@1#thm:clock, X-G-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 satisfies

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

These identities hold on all operators and after adjoining every untouched reference. The measured clock names are consumed; no quantum clock factor is returned.

Proof. The recorded Kraus maps are $∣ j ⟩_{R} \otimes D_{j}$ . Their squared moduli sum to $\sum_{j} P_{j} \otimes I_{A} = I$ . In a clock-basis expansion $X = \sum_{j, l} ∣ τ_{j} ⟩ ⟨ τ_{l} ∣ \otimes X_{j l}$ , both recorded $j$ -blocks equal $X_{j j}$ . Their sum is the partial trace. The same calculation includes an arbitrary reference factor. Consumption is the specified destructive interface, whose quantum codomain is $A$ , not the measured clock.

Displayed formulas in this passage

X-G-ST@1#eq:clock · line 2216
$M (X) = j \sum ∣ j ⟩ ⟨ j ∣_{R} \otimes D_{j} X D_{j}^{†} = j \sum ∣ j ⟩ ⟨ j ∣_{R} \otimes Tr_{C_{clk}} [(P_{j} \otimes I_{A}) X (P_{j} \otimes I_{A})]$
unlabeled · line 2225
$Tr_{R} M (X) = Tr_{C_{clk}} X .$

Theorem: Complete clock instrument

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