Manuscript VI-B@1 · VI-B-v1.tex · theorem · 33798723

Theorem: Complete destructive clock and scored instrument

Destructive clock, complete instrument, and dictionaries · lines 1578–1625 · proof included or linked

VI-B@1#thm:instrument, VI-B@1#eq:clock, VI-B@1#eq:instrument

The clock channel is

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

The subsequent code-selected table readout is

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

These maps are completely positive and trace preserving on all input operators, including every untouched reference extension.

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

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

The measured clock names are consumed; they are not quantum outputs of either instrument.

Proof. For (eq:clock), use Kraus operators $∣ k ⟩_{R} \otimes D_{k}$ . Their squared moduli sum to $\sum_{k} P_{k} \otimes I_{X} = I_{C X}$ . Expansion in the clock basis gives the partial-trace identity.

For (eq:instrument), use the recorded Kraus operators

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

Since $\sum_{y} M_{k, y}^{†} M_{k, y} = I_{X}$ , their squared moduli also sum to $I_{C X}$ . Tensoring these operator identities with any reference identity proves the complete assertions. The conditional score follows from (eq:score). The specified implementation consumes $C$ , and its output type contains $R, Y, X$ , not a quantum $C$ .

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VI-B@1#eq:clock · line 1581
$M_{C} (Z) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} Z D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{X}) Z (P_{k} \otimes I_{X})] .$
VI-B@1#eq:instrument · line 1589
$I (Z) = k, y \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes M_{k, y} D_{k} Z D_{k}^{†} M_{k, y}^{†} .$
unlabeled · line 1600
$E [B y ∣ k] = \frac{Tr ( A _{k} σ _{k} )}{p _{k}} .$
unlabeled · line 1615
$L_{k, y} = ∣ k ⟩_{R} \otimes ∣ y ⟩_{Y} \otimes M_{k, y} D_{k} .$

Theorem: Complete destructive clock and scored instrument

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