Manuscript X-E-ST@1 · X-E-ST.tex · theorem · 44614221

Theorem: Complete instrument, including references

Finite supplied evaluations and the Enter instrument › Complete destructive clock · lines 2296–2330 · proof included or linked

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

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The map

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

is CPTP and satisfies

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

Both identities hold after adjoining any untouched reference. All measured clock names are consumed; no quantum clock output is returned.

Proof. The Kraus maps are $∣ k ⟩_{R} \otimes D_{k}$ , and

k \sum D_{k}^{†} D_{k} = k \sum P_{k} \otimes I_{A} = I .

Expand $X = \sum_{j, l} ∣ τ_{j} ⟩ ⟨ τ_{l} ∣ \otimes X_{j l}$ . Both expressions in the $k$ -th recorded block equal $X_{k k}$ ; their sum is the partial trace. The same calculation includes any reference factor. Consumption is the declared destructive-readout interface: its quantum codomain contains $A$ , not $C_{clk}$ .

Displayed formulas in this passage

X-E-ST@1#eq:clock · line 2299
$M (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C_{clk}} [(P_{k} \otimes I_{A}) X (P_{k} \otimes I_{A})]$
unlabeled · line 2308
$Tr_{R} M (X) = Tr_{C_{clk}} X .$
unlabeled · line 2318
$k \sum D_{k}^{†} D_{k} = k \sum P_{k} \otimes I_{A} = I .$

Theorem: Complete instrument, including references

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