Manuscript Hawk05-C@1 · Hawk05-C.tex · proposition · ba0719f1

Proposition: The gap is marginal coherence

Nonuniform cells: an exact temporal proxy › What the entropy gap actually means · lines 690–726 · proof included or linked

Hawk05-C@1#prop:coherence, Hawk05-C@1#eq:coherencegap

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For every state $ρ_{R}$ ,

H (Y) - S (ρ_{R}) = S (Δ_{R} ρ_{R}) - S (ρ_{R}) = D_{2} (ρ_{R} ∥ Δ_{R} ρ_{R}) \geq 0.

For the Haar proxy,

L (r, b) - P (r, b) = ⎩ ⎨ ⎧ \frac{r - 1}{2 b ln 2}, \frac{H _{r} - H _{b} + ( b - 1 ) / ( 2 r )}{ln 2}, r \leq b, r \geq b .

This is not a whole-code recovery diagnostic.

Proof. The diagonal logarithm satisfies $Tr (ρ_{R} lo g_{2} Δ_{R} ρ_{R}) = Tr (Δ_{R} ρ_{R} lo g_{2} Δ_{R} ρ_{R})$ . Positivity ensures the required support containment. The relative-entropy identity and nonnegativity follow. Subtract (eq:PageMean) from (eq:RecordMean) for the two cases.

For a Bell input to a $d$ -dimensional identity channel, the accessible marginal is $I / d$ , so the marginal gap is zero. Nevertheless, destructive basis readout has recovery error $b_{d}$ . Thus a vanishing gap does not mean no quantum-recovery loss.

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Hawk05-C@1#eq:coherencegap · line 701
$L (r, b) - P (r, b) = ⎩ ⎨ ⎧ \frac{r - 1}{2 b ln 2}, \frac{H _{r} - H _{b} + ( b - 1 ) / ( 2 r )}{ln 2}, r \leq b, r \geq b .$
unlabeled · line 693
$H (Y) - S (ρ_{R}) = S (Δ_{R} ρ_{R}) - S (ρ_{R}) = D_{2} (ρ_{R} ∥ Δ_{R} ρ_{R}) \geq 0.$

Proposition: The gap is marginal coherence

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