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

Special cases are not the thermal theorem

Stage 4: change one hypothesis at a time › Special cases are not the thermal theorem · lines 1602–1631

The one-use proof gives the following exact specializations:

γ = 0 : γ = 1 : ν = 0 : ν = 1/2 : A_{γ, ν} = id, p max F_{γ, ν} (p) = 1, p^{⋆} = 1/2; A_{γ, ν} (X) = Tr (X) diag (1 - ν, ν), χ = 0; F_{γ, 0} (p) = h_{2} ((1 - γ) p) - g_{2} (γ (1 - γ) p^{2}); p max F_{γ, 1/2} (p) = 1 - h_{2} (\frac{1 + 1 - γ}{2}) .

For the unital case with $γ < 1$ , symmetry around $1/2$ and strict concavity identify $p^{⋆} = 1/2$ .

Unital qubit channels have an independent additivity theory King; the symmetric GAD channel also has an explicit classical-channel treatment SiddhuEtAl. Those special regimes do not establish every thermal parameter or every claimed entropy inequality.

Displayed formulas in this passage

unlabeled · line 1605
$γ = 0 : γ = 1 : ν = 0 : ν = 1/2 : A_{γ, ν} = id, p max F_{γ, ν} (p) = 1, p^{⋆} = 1/2; A_{γ, ν} (X) = Tr (X) diag (1 - ν, ν), χ = 0; F_{γ, 0} (p) = h_{2} ((1 - γ) p) - g_{2} (γ (1 - γ) p^{2}); p max F_{γ, 1/2} (p) = 1 - h_{2} (\frac{1 + 1 - γ}{2}) .$