Manuscript Oai265-A@2 · Oai265-A.tex · proposition · c0a34a45

Proposition: Thermal spectra do not repair the inference

Stage 5: complete channels and the entropy separation › The sharp replacement-channel benchmark · lines 1441–1472 · proof included or linked

Oai265-A@2#prop:thermal

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

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

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

For every pure input, the accessible spectra coincide and both entropies equal $S (σ_{K})$ . If $H_{R} = I_{C_{R}} \otimes H_{K}$ , all accessible energy-measurement distributions coincide as well. The optimal recovery errors are still $0$ and $1 - d^{- 2}$ .

Proof. The accessible channels are

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

A pure logical factor changes neither the nonzero spectrum nor the entropy of $σ_{K}$ . Spectral effects of $H_{R}$ act only on $K$ . Tracing $K$ decodes the first channel, and the second is constant.

Displayed formulas in this passage

unlabeled · line 1446
$V_{return} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣0 ⟩_{L},$
unlabeled · line 1450
$V_{retain} ∣ ψ ⟩ = ∣0 ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L} .$
unlabeled · line 1463
$X ⟼ X \otimes σ_{K}, X ⟼ Tr (X) ∣0 ⟩ ⟨ 0∣ \otimes σ_{K} .$

Proposition: Thermal spectra do not repair the inference

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