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

The external claims, not local premises

Stage 0: freeze the channel, units, and claims › The external claims, not local premises · lines 319–369

Oai276-A@1#eq:objective, Oai276-A@1#rec:main, Oai276-A@1#rec:partner

Set

v_{γ, ν} (p) = γ ν (1 - ν) + γ (1 - γ) (p - ν)^{2}, F_{γ, ν} (p) = h_{2} ((1 - γ) p + γ ν) - g_{2} (v_{γ, ν} (p)) .

Family 276 asserts, for every $(γ, ν) \in [0, 1]^{2}$ and $n \geq 1$ ,

χ (A_{γ, ν}^{\otimes n}) = n 0 \leq p \leq 1 max F_{γ, ν} (p), C (A_{γ, ν}) = χ (A_{γ, ν}) .

It further asserts that every maximizing $p$ gives attaining equiprobable phase-pair products, where

ϕ_{\pm} (p) = 1 - p ∣0 ⟩ \pm p ∣1 ⟩ .

The maximum is one-dimensional; this is not a closed algebraic evaluation of its maximizer OpenAI276.

The source separately asserts, for every finite-dimensional CPTP partner $N$ ,

χ (A \otimes N) = χ (A) + χ (N),

S_{m i n} (A \otimes N) = S_{m i n} (A) + S_{m i n} (N), C (A \otimes N) = C (A) + C (N) .

The partner retains its own regularization. This quantifier is stronger than repeated uses of $A$ , and it is outside the inspected Lean scope OpenAI276,LeanScope276.

Displayed formulas in this passage

Oai276-A@1#eq:objective · line 322
$v_{γ, ν} (p) = γ ν (1 - ν) + γ (1 - γ) (p - ν)^{2}, F_{γ, ν} (p) = h_{2} ((1 - γ) p + γ ν) - g_{2} (v_{γ, ν} (p)) .$
unlabeled · line 337
$χ (A_{γ, ν}^{\otimes n}) = n 0 \leq p \leq 1 max F_{γ, ν} (p), C (A_{γ, ν}) = χ (A_{γ, ν}) .$
unlabeled · line 346
$ϕ_{\pm} (p) = 1 - p ∣0 ⟩ \pm p ∣1 ⟩ .$
unlabeled · line 357
$χ (A \otimes N) = χ (A) + χ (N),$
unlabeled · line 360
$S_{m i n} (A \otimes N) = S_{m i n} (A) + S_{m i n} (N), C (A \otimes N) = C (A) + C (N) .$

The external claims, not local premises

Displayed formulas in this passage

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