Manuscript Hawk04-B@2 · Hawk04-B.tex · proposition · ba3b0713

Proposition: A complete readout dilation retains omitted information

Residual readout, local forgetting, and the apparatus › The coherent apparatus ledger · lines 1109–1132 · proof included or linked

Hawk04-B@2#prop:measurement-dilation, Hawk04-B@2#eq:measurement-isometry

Back to manuscript

For the destructive instrument (eq:Enter), define

W_{read} ∣ ψ ⟩ = x \sum ∣ x ⟩_{K} ∣ x ⟩_{E} \otimes M_{x} ∣ ψ ⟩ .

This is an isometry. Tracing $E$ gives the observable hybrid readout instrument. The complete output $K E T$ preserves the input globally, whereas the classical interface $K$ , by itself, generally does not.

Proof. The identities $⟨ x, y ⟩ = δ_{x y}$ and $\sum_{x} M_{x}^{†} M_{x} = I$ give $W_{read}^{†} W_{read} = I$ . Tracing the orthogonal $E$ -labels removes cross-outcome terms and yields (eq:Enter). Complete recoverability follows from Theorem thm:global.

Displayed formulas in this passage

Hawk04-B@2#eq:measurement-isometry · line 1112
$W_{read} ∣ ψ ⟩ = x \sum ∣ x ⟩_{K} ∣ x ⟩_{E} \otimes M_{x} ∣ ψ ⟩ .$

Nearby in this section

section: The coherent apparatus ledger