Manuscript Oai272-A@1 · Oai272-A.tex · proposition · a40828a0

Proposition: Selected data do not identify the relevant channel

Stage 5: secrecy, recovery, and gravity remain separate · lines 1926–1945 · proof included or linked

Oai272-A@1#prop:selected

For $C ≃ R ≃ B ≃ C^{d}$ , the isometries

V_{0} ∣ ψ ⟩ = ∣ ψ ⟩_{R} ∣0 ⟩_{B}, V_{1} ∣ ψ ⟩ = ∣0 ⟩_{R} ∣ ψ ⟩_{B}

have identical accessible pure-input spectra, all with entropy zero. The first accessible channel exactly recovers the logical state; the second is constant and cannot do so.

Proof. Their accessible channels are $id_{C}$ and $X \mapsto Tr (X) ∣0 ⟩ ⟨ 0∣$ . Every pure accessible output has spectrum $(1, 0, \dots, 0)$ . The first is inverted by identification of $R, C$ ; every postprocessing of the second is constant.

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unlabeled · line 1929
$V_{0} ∣ ψ ⟩ = ∣ ψ ⟩_{R} ∣0 ⟩_{B}, V_{1} ∣ ψ ⟩ = ∣0 ⟩_{R} ∣ ψ ⟩_{B}$

Proposition: Selected data do not identify the relevant channel

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