Manuscript Oai211-A@1 · Oai211-A.tex · section · 3f3e4f63

Selected marginals do not specify the complete channel

Metric convergence and quantum recovery are separate › Selected marginals do not specify the complete channel · lines 1694–1714

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For any fixed selected state $τ_{S}$ , choose a purification $χ_{S B_{0}}$ . The two isometries

ψ ⟼ χ_{S B_{0}} \otimes ψ_{R_{1}} \otimes ∣ 0 ⟩_{B_{1}}, ψ ⟼ χ_{S B_{0}} \otimes ∣ 0 ⟩_{R_{1}} \otimes ψ_{B_{1}}

have the same selected $S$ marginal for every input, but opposite full accessible recovery behavior. This applies even when $τ_{S}$ is a chosen thermal density. It does not construct a Hawking state.

Cheung's En07-C explicitly separates a thermal input from a complete complementary-channel verdict CheungEn07C. The selected packet and apparatus distinctions in Em03-A–C likewise do not supply an identified quantum channel on the present map CheungEm03ABC.

Displayed formulas in this passage

unlabeled · line 1699
$ψ ⟼ χ_{S B_{0}} \otimes ψ_{R_{1}} \otimes ∣ 0 ⟩_{B_{1}}, ψ ⟼ χ_{S B_{0}} \otimes ∣ 0 ⟩_{R_{1}} \otimes ψ_{B_{1}}$