Manuscript Hawk05-B@1 · Hawk05-B.tex · proposition · f2e5d4f5

Proposition: Selected spectra and thermal marginals

Explicit entropy–recovery separations › Same entropy history, inaccessible logical memory · lines 1191–1213 · proof included or linked

Hawk05-B@1#prop:thermal

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Let $σ_{K}$ have purification $∣ χ ⟩_{K P}$ , and put $R = C_{R} K$ , $B = P L$ , with $L ≃ C$ . The isometries

V_{good} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ 0 ⟩_{L}, V_{bad} ∣ ψ ⟩ = ∣ 0 ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L}

have identical accessible spectra for every pure input, but recovery errors $0$ and $a_{d}$ . For $H_{R} = I_{C_{R}} \otimes H_{K}$ , their selected energy laws agree. One may take $σ_{K}$ to be Gibbs.

Proof. The accessible channels are $X \mapsto X \otimes σ_{K}$ and $X \mapsto Tr (X) ∣ 0 ⟩ ⟨ 0 ∣ \otimes σ_{K}$ . The spectrum, recovery, and energy conclusions follow directly.

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unlabeled · line 1196
$V_{good} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ 0 ⟩_{L}, V_{bad} ∣ ψ ⟩ = ∣ 0 ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L}$

Proposition: Selected spectra and thermal marginals

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