Manuscript VII-A-ST@1 · VII-A-ST.tex · theorem · c39a67a3

Theorem: Complete destructive readout

Finite encoding and the destructive-clock instrument › Clock and complete recorded map · lines 1608–1655 · proof included or linked

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

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The clock channel is

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

The clock-selected table instrument is

I (Z) = s, y \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes M_{s, y} D_{s} Z D_{s}^{†} M_{s, y}^{†} .

Both maps are completely positive and trace preserving, including every untouched reference extension.

For $σ_{s} = D_{s} ρ D_{s}^{†}$ and $p_{s} = Tr σ_{s} > 0$ ,

E [Λ y ∣ s] = \frac{Tr ( A _{s} σ _{s} )}{p _{s}} .

The measured clock names are consumed. No quantum clock factor is returned.

Proof. The clock Kraus operators are $∣ s ⟩_{R} \otimes D_{s}$ , and their squared moduli sum to $\sum_{s} P_{s} \otimes I_{X} = I_{C X}$ . Expanding $Z$ in the $τ$ -basis proves the partial-trace expression.

For the full instrument use

L_{s, y} = ∣ s ⟩_{R} \otimes ∣ y ⟩_{Y} \otimes M_{s, y} D_{s} .

Equation (eq:scoreidentity) gives $\sum_{s, y} L_{s, y}^{†} L_{s, y} = I_{C X}$ . These are operator identities, so tensoring with any reference identity proves the complete statements. The score formula follows from (eq:scoreidentity). The output interface contains $R, Y, X$ , not quantum $C$ .

Displayed formulas in this passage

VII-A-ST@1#eq:clock · line 1611
$M_{C} (Z) = s \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes D_{s} Z D_{s}^{†} = s \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes Tr_{C} [(P_{s} \otimes I_{X}) Z (P_{s} \otimes I_{X})] .$
VII-A-ST@1#eq:instrument · line 1619
$I (Z) = s, y \sum ∣ s ⟩ ⟨ s ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes M_{s, y} D_{s} Z D_{s}^{†} M_{s, y}^{†} .$
unlabeled · line 1630
$E [Λ y ∣ s] = \frac{Tr ( A _{s} σ _{s} )}{p _{s}} .$
unlabeled · line 1646
$L_{s, y} = ∣ s ⟩_{R} \otimes ∣ y ⟩_{Y} \otimes M_{s, y} D_{s} .$

Theorem: Complete destructive readout

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