Manuscript I-B@1 · I-B-v1.tex · theorem · 5357688f

Theorem: Complete destructive clock instrument

Finite names, access, control, and destructive Enter › The clock instrument · lines 450–492 · proof included or linked

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

The specified readout has channel

M_{C} (Z) = k = 0 \sum^{D - 1} ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} Z D_{k}^{†} = k = 0 \sum^{D - 1} ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{X}) Z (P_{k} \otimes I_{X})] .

It is completely positive and trace preserving. Its $k$ -th subnormalized matter block is

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

For $p_{k} > 0$ , the conditional matter state is $σ_{k} / p_{k}$ . No normalized state is assigned to a zero-probability outcome. Every identity holds after tensoring with $id_{E}$ .

Proof. After applying $B_{C}$ , the destructive measurement bra for outcome $k$ is

(⟨ k ∣ B_{C}) \otimes I_{X} = D_{k} .

The Kraus operator including the classical output is

L_{k} = ∣ k ⟩_{R} \otimes D_{k} .

Consequently the channel is (eq:instrument), and

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

This proves complete positivity and trace preservation. The partial-trace expression follows by expanding in the clock basis. The block probabilities and conditional states are the specified Born semantics. Tensoring each operator identity with $I_{E}$ proves the reference versions.

Displayed formulas in this passage

I-B@1#eq:instrument · line 453
$M_{C} (Z) = k = 0 \sum^{D - 1} ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} Z D_{k}^{†} = k = 0 \sum^{D - 1} ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{X}) Z (P_{k} \otimes I_{X})] .$
unlabeled · line 463
$σ_{k} = D_{k} ρ D_{k}^{†}, p_{k} = Tr σ_{k} .$
unlabeled · line 475
$(⟨ k ∣ B_{C}) \otimes I_{X} = D_{k} .$
unlabeled · line 479
$L_{k} = ∣ k ⟩_{R} \otimes D_{k} .$
unlabeled · line 483
$k \sum L_{k}^{†} L_{k} = k \sum P_{k} \otimes I_{X} = I_{C X} .$

Theorem: Complete destructive clock instrument

Displayed formulas in this passage

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