Manuscript Hawk05-A@1 · Hawk05-A.tex · theorem · 449f85bb

Theorem: Selected spectral and thermal separation

Entropy and thermal observations do not certify recovery · lines 1273–1311 · proof included or linked

Hawk05-A@1#thm:spectra

Fix a state $σ_{K}$ , a purification $∣ χ ⟩_{K P}$ , and $L ≃ C$ . Use the same output spaces

A = C_{A} \otimes K, E = P \otimes L .

The isometries

V_{return} ∣ ψ ⟩ V_{retain} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{A}} ∣ χ ⟩_{K P} ∣ 0 ⟩_{L}, = ∣ 0 ⟩_{C_{A}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L}

have, for every pure logical input, identical accessible spectra and entropy $S (σ_{K})$ , but

ε_{rec} (N_{return}) = 0, ε_{rec} (N_{retain}) = a_{d} .

For $H_{A} = I_{C_{A}} \otimes H_{K}$ , their energy-measurement laws also agree for every logical input. One may choose $σ_{K}$ Gibbs.

Proof. Their accessible channels are

X ⟼ X \otimes σ_{K}, X ⟼ Tr (X) ∣ 0 ⟩ ⟨ 0 ∣ \otimes σ_{K} .

For pure $X$ , both outputs have the nonzero spectrum of $σ_{K}$ . Tracing $K$ decodes the first channel. The second is replacement, so Lemma lem:replacement applies. Every spectral effect of $H_{A}$ acts trivially on $C_{A}$ , proving equality of the selected energy laws.

Displayed formulas in this passage

unlabeled · line 1281
$V_{return} ∣ ψ ⟩ V_{retain} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{A}} ∣ χ ⟩_{K P} ∣ 0 ⟩_{L}, = ∣ 0 ⟩_{C_{A}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L}$
unlabeled · line 1277
$A = C_{A} \otimes K, E = P \otimes L .$
unlabeled · line 1291
$ε_{rec} (N_{return}) = 0, ε_{rec} (N_{retain}) = a_{d} .$
unlabeled · line 1301
$X ⟼ X \otimes σ_{K}, X ⟼ Tr (X) ∣ 0 ⟩ ⟨ 0 ∣ \otimes σ_{K} .$

Theorem: Selected spectral and thermal separation

Displayed formulas in this passage

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