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

Gate 3: the channel substitution

Stage 1: audit the five external proof interfaces › Gate 3: the channel substitution · lines 695–744

Oai276-A@1#eq:coefficients

The substitution itself is elementary and can be checked without assuming (eq:audit-thermal). Put

a = 1 - γ ν, b = γ (1 - ν), c = γ ν, d = 1 - γ (1 - ν), k = 1 - γ .

Then

a + c = b + d = 1, a d - b c = k^{2} .

For a normalized pure input

ψ = 1 - p ∣0 ⟩ \otimes ψ_{0} + p ∣1 ⟩ \otimes ψ_{1}

and a CPTP partner with Kraus matrices $V_{ℓ}$ , form rectangular matrices with columns

X_{ℓ} = V_{ℓ} ψ_{0}, Y_{ℓ} = V_{ℓ} ψ_{1} .

Their common column index gives

X X^{†} = N (∣ ψ_{0} ⟩ ⟨ ψ_{0} ∣), Y Y^{†} = N (∣ ψ_{1} ⟩ ⟨ ψ_{1} ∣), X Y^{†} = N (∣ ψ_{0} ⟩ ⟨ ψ_{1} ∣) .

No orthogonality of $ψ_{0}, ψ_{1}$ is required.

The thermal coefficients are

α = a (1 - p), β = b p, η = c (1 - p), δ = d p, K = k p (1 - p) .

Equation (eq:coefficients) verifies their required identity, and their scalar $u$ is $v_{γ, ν} (p)$ .

The algebra therefore identifies exactly where the thermal entropy theorem would enter. It does not prove that theorem. In particular, a Kraus representation is not a decoder.

Displayed formulas in this passage

Oai276-A@1#eq:coefficients · line 707
$a + c = b + d = 1, a d - b c = k^{2} .$
unlabeled · line 699
$a = 1 - γ ν, b = γ (1 - ν), c = γ ν, d = 1 - γ (1 - ν), k = 1 - γ .$
unlabeled · line 713
$ψ = 1 - p ∣0 ⟩ \otimes ψ_{0} + p ∣1 ⟩ \otimes ψ_{1}$
unlabeled · line 719
$X_{ℓ} = V_{ℓ} ψ_{0}, Y_{ℓ} = V_{ℓ} ψ_{1} .$
unlabeled · line 723
$X X^{†} = N (∣ ψ_{0} ⟩ ⟨ ψ_{0} ∣), Y Y^{†} = N (∣ ψ_{1} ⟩ ⟨ ψ_{1} ∣), X Y^{†} = N (∣ ψ_{0} ⟩ ⟨ ψ_{1} ∣) .$
unlabeled · line 731
$α = a (1 - p), β = b p, η = c (1 - p), δ = d p, K = k p (1 - p) .$