Manuscript VII-E-ST@1 · VII-E-ST.tex · corollary · d28d9c5d

Corollary: An informative classical record prevents full recovery

Recovery obstruction, deterministic packets, and balance · lines 1473–1496 · proof included or linked

VII-E-ST@1#cor:informative

For an instrument with recorded channel

N (X) = w \sum ∣ w ⟩ ⟨ w ∣_{R} \otimes Ψ_{w} (X),

if some effect $E_{w} = Ψ_{w}^{*} (I)$ is not scalar, no channel recovers every input from the entire recorded output, including its residual.

Proof. Choose branch Kraus operators $K_{w, ν}$ . A dilation can duplicate the classical outcome into its environment:

V ψ = w, ν \sum ∣ w ⟩_{R} \otimes K_{w, ν} ψ \otimes ∣ w, ν ⟩_{E} .

The environmental marginal has outcome- $w$ probability $Tr (E_{w} ρ)$ . A constant complement would make this probability independent of every input, requiring $E_{w} = p_{w} I$ . Apply Theorem thm:decouple.

Displayed formulas in this passage

unlabeled · line 1476
$N (X) = w \sum ∣ w ⟩ ⟨ w ∣_{R} \otimes Ψ_{w} (X),$
unlabeled · line 1488
$V ψ = w, ν \sum ∣ w ⟩_{R} \otimes K_{w, ν} ψ \otimes ∣ w, ν ⟩_{E} .$

Corollary: An informative classical record prevents full recovery

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