Manuscript Hawk07-B@1 · Hawk07-B.tex · section · b9785538

The album prediction

Entropy predictions and a genuinely scaled turnover theorem › The album prediction · lines 1247–1275

Hawk07-B@1#eq:album

For destructive readout in a prescribed basis,

(Enter_{A} \otimes id_{T}) (X) = y \sum ∣ y ⟩ ⟨ y ∣_{Y} \otimes (⟨ y ∣_{A} \otimes I_{T}) X (∣ y ⟩_{A} \otimes I_{T}) .

The measured name is consumed; $T$ survives.

For every fixed process with an append-only record,

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

This is the classical chain rule, including adaptive readout. It concerns the unedited record of one process, not a selected subsequence or a reset memory.

Thus coherent entropy may rise and fall while the album cannot. Neither curve determines recovery by itself CheungHawk05ABC. In the nested model, leakage and optimal recovery error remain nonincreasing through the coherent peak.

Displayed formulas in this passage

Hawk07-B@1#eq:album · line 1259
$H (Y_{\leq k + 1}) = H (Y_{\leq k}) + H (Y_{k + 1} ∣ Y_{\leq k}) \geq H (Y_{\leq k}) .$
unlabeled · line 1250
$(Enter_{A} \otimes id_{T}) (X) = y \sum ∣ y ⟩ ⟨ y ∣_{Y} \otimes (⟨ y ∣_{A} \otimes I_{T}) X (∣ y ⟩_{A} \otimes I_{T}) .$