Manuscript Hawk05-B@1 · Hawk05-B.tex · section · f2e5d4f5

Destructive Enter gives a different entropy history

Destructive Enter gives a different entropy history · lines 763–789

For a register $A$ with basis ${∣ y ⟩}$ , define destructive readout by

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

The returned quantum names are those of $K$ , not $A$ . A complete dilation is

W_{read} ∣ y ⟩_{A} = ∣ y ⟩_{Y} ∣ y ⟩_{L},

where $L$ is unreturned. The complete evolution remains isometric; the returned classical record alone is dephased.

Finite adaptive compositions are CPTP: composing the unnormalized path maps $K_{h}$ gives $\sum_{h} K_{h}^{†} K_{h} = I$ . This is the ordinary finite instrument setting also used in Cheung's hybrid and sequential interfaces CheungVIIG,CheungEn02C.

Displayed formulas in this passage

unlabeled · line 768
$(Enter_{A} \otimes id_{K}) (X) = y \sum ∣ y ⟩ ⟨ y ∣_{Y} \otimes (⟨ y ∣_{A} \otimes I_{K}) X (∣ y ⟩_{A} \otimes I_{K}) .$
unlabeled · line 776
$W_{read} ∣ y ⟩_{A} = ∣ y ⟩_{Y} ∣ y ⟩_{L},$