Manuscript Oai290-A@1 · Oai290-A.tex · proposition · 7c6e104a

Proposition: Selected marginal nonidentification

Analytic examples and separation tests › Identical selected thermal data with opposite recovery · lines 1796–1823 · proof included or linked

Oai290-A@1#prop:thermal

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For any finite state $τ_{S}$ , there are complete isometries with the same selected accessible marginal $τ_{S}$ for every input, but with full accessible optimal recovery errors respectively $0$ and $1 - d^{- 2}$ .

Proof. Choose a purification $χ_{S B_{0}}$ of $τ_{S}$ . Let the full accessible output be $A = S \otimes A_{1}$ , with $A_{1} ≅ C$ , and take $B = B_{0} \otimes B_{1}$ , $B_{1} ≅ C$ . Define

V_{good} ψ = χ_{S B_{0}} \otimes ψ_{A_{1}} \otimes ∣ 0 ⟩_{B_{1}},

V_{bad} ψ = χ_{S B_{0}} \otimes ∣ 0 ⟩_{A_{1}} \otimes ψ_{B_{1}} .

Both are isometries and have selected $S$ marginal $τ_{S}$ . The good channel returns the entire input in $A_{1}$ . The bad full accessible channel is replacement. Apply Theorem thm:exact and Proposition prop:classical.

Displayed formulas in this passage

unlabeled · line 1810
$V_{good} ψ = χ_{S B_{0}} \otimes ψ_{A_{1}} \otimes ∣ 0 ⟩_{B_{1}},$
unlabeled · line 1814
$V_{bad} ψ = χ_{S B_{0}} \otimes ∣ 0 ⟩_{A_{1}} \otimes ψ_{B_{1}} .$

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section: Identical selected thermal data with opposite recovery