Manuscript V-B@1 · V-B-v1.tex · theorem · 3fadc1ad

Theorem: Complete clock and response instrument

Finite compression and the complete Enter interface › Destructive clock readout · lines 1783–1829 · proof included or linked

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

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The destructive 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})] .

Its composition with the code-selected response reading 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}^{†} .

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

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.

Proof. The recorded Kraus operators for (eq:clock) are $∣ k ⟩_{R} \otimes D_{k}$ , with squared sum $I_{C X}$ . Rank-one clock projection gives the partial-trace identity.

For (eq:instrument), use the Kraus family $∣ k ⟩_{R} \otimes ∣ y ⟩_{Y} \otimes M_{k, y} D_{k}$ . Since $\sum_{y} M_{k, y}^{†} M_{k, y} = I$ , its squared sum is again $I_{C X}$ . Tensoring these identities with any reference identity proves the complete assertions.

Finally $\sum_{y} B y E_{k, y} = A_{k}$ . The destructive implementation changes the clock input into a classical code; its quantum output contains $X$ , not $C$ . No normalized conditional state is assigned when $p_{k} = 0$ .

Displayed formulas in this passage

V-B@1#eq:clock · line 1786
$M_{C} (Z) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} Z D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{X}) Z (P_{k} \otimes I_{X})] .$
V-B@1#eq:instrument · line 1794
$I (Z) = k, y \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes M_{k, y} D_{k} Z D_{k}^{†} M_{k, y}^{†} .$
unlabeled · line 1806
$E [B y ∣ k] = \frac{Tr ( A _{k} σ _{k} )}{p _{k}} .$

Theorem: Complete clock and response instrument

Displayed formulas in this passage

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