Manuscript Oai265-A@2 · Oai265-A.tex · proposition · c0a34a45

Proposition: Identical pure-input entropy and different recovery

Stage 5: complete channels and the entropy separation › The sharp replacement-channel benchmark · lines 1407–1432 · proof included or linked

Oai265-A@2#prop:goodbad

With $R ≃ B ≃ C ≃ C^{d}$ , define

V_{good} ∣ ψ ⟩ = ∣ ψ ⟩_{R} ∣0 ⟩_{B}, V_{bad} ∣ ψ ⟩ = ∣0 ⟩_{R} ∣ ψ ⟩_{B} .

For every pure logical input, both accessible states have entropy zero and the same spectrum. Nevertheless,

ε_{rec} (N_{good}) = δ_{leak} (E_{good}) = 0,

whereas

ε_{rec} (N_{bad}) = δ_{leak} (E_{bad}) = 1 - d^{- 2} .

Proof. The accessible channels are identity and replacement; the complements are replacement and identity. Apply Lemma lem:constant.

Displayed formulas in this passage

unlabeled · line 1410
$V_{good} ∣ ψ ⟩ = ∣ ψ ⟩_{R} ∣0 ⟩_{B}, V_{bad} ∣ ψ ⟩ = ∣0 ⟩_{R} ∣ ψ ⟩_{B} .$
unlabeled · line 1417
$ε_{rec} (N_{good}) = δ_{leak} (E_{good}) = 0,$
unlabeled · line 1422
$ε_{rec} (N_{bad}) = δ_{leak} (E_{bad}) = 1 - d^{- 2} .$