Manuscript En08-B@1 · En08-B.tex · proposition · 23cac52a

Proposition: Record-only quantum recovery obstruction

Failure tests and finite-scale attainment › Record-only transmission · lines 1340–1370 · proof included or linked

En08-B@1#prop:record, En08-B@1#eq:record-bound

If a payload is converted into a finite classical record and a decoder prepares a $D$ -dimensional quantum output from that record, then the resulting channel $Λ$ satisfies

d_{⋄} (Λ, id_{D}) \geq 1 - \frac{1}{D} .

Proof. The composite channel has measure-and-prepare form

Λ (Z) = a \sum Tr (F_{a} Z) ω_{a}, F_{a} \geq 0, a \sum F_{a} = I .

Its output on half of $∣ Ω_{D} ⟩ ⟨ Ω_{D} ∣$ is separable between reference and output.

For unit product vectors,

∣ ⟨ Ω_{D} ∣ u \otimes v ⟩ ∣^{2} \leq \frac{1}{D}

by Cauchy–Schwarz. Convexity gives the same overlap bound for every separable state. Testing the Bell projector yields trace distance at least $1 - 1/ D$ from $∣ Ω_{D} ⟩ ⟨ Ω_{D} ∣$ . This is an admissible referenced channel test and proves (eq:record-bound).

Displayed formulas in this passage

En08-B@1#eq:record-bound · line 1345
$d_{⋄} (Λ, id_{D}) \geq 1 - \frac{1}{D} .$
unlabeled · line 1353
$Λ (Z) = a \sum Tr (F_{a} Z) ω_{a}, F_{a} \geq 0, a \sum F_{a} = I .$
unlabeled · line 1361
$∣ ⟨ Ω_{D} ∣ u \otimes v ⟩ ∣^{2} \leq \frac{1}{D}$

Proposition: Record-only quantum recovery obstruction

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