Manuscript II-B@1 · II-B-v1.tex · theorem · bfd2eb3c

Theorem: Complete destructive clock instrument

Finite names and the destructive clock › The clock instrument · lines 1307–1338 · proof included or linked

II-B@1#thm:instrument, II-B@1#eq:instrument

The instrument is

M_{C} (Z) = k = 0 \sum^{3} ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} Z D_{k}^{†} .

It is completely positive and trace preserving. On an input state $ρ$ , its subnormalized blocks are

σ_{k} = D_{k} ρ D_{k}^{†}, p_{k} = Tr σ_{k} .

For $p_{k} > 0$ , the conditional state is $σ_{k} / p_{k}$ . All identities hold after tensoring with $id_{E}$ . The clock factor $C$ is absent from the returned quantum context.

Proof. The destructive bra for outcome $k$ is $⟨ k ∣ B_{C} = ⟨ τ_{k} ∣$ . The Kraus operator including its record is $L_{k} = ∣ k ⟩_{R} \otimes D_{k}$ , and

k \sum L_{k}^{†} L_{k} = k \sum P_{k} \otimes I_{X} = I_{C X} .

This proves complete positivity and trace preservation. Tensoring the Kraus operators with $I_{E}$ proves the reference statement. Their codomain contains $X$ , not $C$ .

Displayed formulas in this passage

II-B@1#eq:instrument · line 1310
$M_{C} (Z) = k = 0 \sum^{3} ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} Z D_{k}^{†} .$
unlabeled · line 1317
$σ_{k} = D_{k} ρ D_{k}^{†}, p_{k} = Tr σ_{k} .$
unlabeled · line 1331
$k \sum L_{k}^{†} L_{k} = k \sum P_{k} \otimes I_{X} = I_{C X} .$

Theorem: Complete destructive clock instrument

Displayed formulas in this passage

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