Manuscript Hawk06-A@1 · Hawk06-A.tex · proposition · c59fe67c

Proposition: Append-only record and the coherence gap

The coherent and destructive entropy calculations › The classical record never turns over · lines 1152–1187 · proof included or linked

Hawk06-A@1#prop:record

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For each fixed unitary and each prescribed append-only destructive record process,

H (Y_{\leq k + 1}) = H (Y_{\leq k}) + H (Y_{k + 1} ∣ Y_{\leq k}) \geq H (Y_{\leq k}) .

For fixed-basis Enter on $R_{k}$ ,

H (Y_{\leq k}) - S (ρ_{R_{k}}) = D_{2} (ρ_{R_{k}} ∥ Δ_{R_{k}} ρ_{R_{k}}) \geq 0,

where $Δ_{R_{k}}$ dephases that basis. This gap is not whole-code recovery error.

Proof. The first assertion is the classical chain rule and does not require Haar scrambling. For the second, the diagonal probabilities of $Δ ρ$ are the measurement probabilities, and

Tr (ρ lo g_{2} Δ ρ) = Tr (Δ ρ lo g_{2} Δ ρ) .

This gives the relative-entropy identity and its nonnegativity. A Bell input to an identity channel has maximally mixed accessible marginal and hence zero marginal coherence gap, but destructive basis readout of that accessible system loses its Bell entanglement.

Displayed formulas in this passage

unlabeled · line 1156
$H (Y_{\leq k + 1}) = H (Y_{\leq k}) + H (Y_{k + 1} ∣ Y_{\leq k}) \geq H (Y_{\leq k}) .$
unlabeled · line 1163
$H (Y_{\leq k}) - S (ρ_{R_{k}}) = D_{2} (ρ_{R_{k}} ∥ Δ_{R_{k}} ρ_{R_{k}}) \geq 0,$
unlabeled · line 1177
$Tr (ρ lo g_{2} Δ ρ) = Tr (Δ ρ lo g_{2} Δ ρ) .$

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section: The classical record never turns over