Manuscript Oai276-A@1 · Oai276-A.tex · section · 215068af

Gate 5: an arbitrary partner is a stronger quantifier

Stage 1: audit the five external proof interfaces › Gate 5: an arbitrary partner is a stronger quantifier · lines 801–834

Oai276-A@1#eq:partner-branch

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For a pure signal on a damping input and a partner input, the source uses the stronger branch estimate

S ((A \otimes N) (∣ ψ ⟩ ⟨ ψ ∣)) \geq e (p) + (1 - p) S (N (∣ ψ_{0} ⟩ ⟨ ψ_{0} ∣)) + pS (N (∣ ψ_{1} ⟩ ⟨ ψ_{1} ∣)) .

This is the rectangular thermal application, not a consequence of same-channel additivity.

The key ensemble identity is that the conditional partner outputs, weighted by $q_{a} (1 - p_{a})$ and $q_{a} p_{a}$ , have mean equal to the partner marginal of the mean full output. If (eq:partner-branch) is supplied, subtracting conditional entropies from

S (σ) \leq h_{2} (t (p)) + S (σ_{partner})

leaves one damping objective and one partner Holevo expression. The same branch estimate supplies the minimum-output lower bound. For regularized capacity, the partner must still be $N^{\otimes n}$ , not replaced by its one-use value SourcePartners276.

This identifies the proof obligation without importing Record rec:partner. Neither this arbitrary-partner bound nor separate-measurement optimality is part of the finite certificate below.

Displayed formulas in this passage

Oai276-A@1#eq:partner-branch · line 805
$S ((A \otimes N) (∣ ψ ⟩ ⟨ ψ ∣)) \geq e (p) + (1 - p) S (N (∣ ψ_{0} ⟩ ⟨ ψ_{0} ∣)) + pS (N (∣ ψ_{1} ⟩ ⟨ ψ_{1} ∣)) .$
unlabeled · line 820
$S (σ) \leq h_{2} (t (p)) + S (σ_{partner})$