Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · proposition · 39b16f67

Proposition: Spectral underdetermination

Thermal calibration is not a complete emission channel › Equal thermal and entropy diagnostics, opposite recovery · lines 1265–1297 · proof included or linked

Hawk04-D@1#prop:spectral-separation

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Choose a finite state $σ_{K}$ , a purification $∣ χ ⟩_{K P}$ , and $L ≃ C$ . Let $R = C_{R} \otimes K$ , $B = P \otimes L$ , and define

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

Then

N_{return} (X) = X \otimes σ_{K}, N_{retain} (X) = Tr (X) ∣ 0 ⟩ ⟨ 0 ∣ \otimes σ_{K} .

For every pure code input, their accessible spectra and entropies coincide. For $H_{R} = I_{C_{R}} \otimes H_{K}$ , all accessible energy-measurement distributions coincide for every input. Nevertheless,

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

Taking $σ_{K}$ Gibbs makes these selected energy statistics thermal.

Proof. Both maps preserve inner products, and partial traces give the channels. For pure inputs, both outputs have the nonzero spectrum of $σ_{K}$ . Every spectral effect of $H_{R}$ acts identically on the logical factor. Tracing $K$ decodes the return map; the retain map is a replacement, so Lemma lem:benchmarks applies.

Displayed formulas in this passage

unlabeled · line 1270
$V_{return} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ 0 ⟩_{L}, V_{retain} ∣ ψ ⟩ = ∣ 0 ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L} .$
unlabeled · line 1277
$N_{return} (X) = X \otimes σ_{K}, N_{retain} (X) = Tr (X) ∣ 0 ⟩ ⟨ 0 ∣ \otimes σ_{K} .$
unlabeled · line 1284
$ε_{rec} (N_{return}) = 0, ε_{rec} (N_{retain}) = a_{d} .$

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section: Equal thermal and entropy diagnostics, opposite recovery