Manuscript En04-A@1 · En04-A.tex · proposition · bab77211

Proposition: Complete coherent readout dilation

Readout, forgetting, and the complete apparatus › The external residual program · lines 1338–1356 · proof included or linked

En04-A@1#prop:readoutdilation

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For the instrument (eq:enter), define

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

This is an isometry. Tracing $E$ gives the observable destructive readout instrument. The complete output preserves the input globally.

Proof. Completeness of the $M_{x}$ gives $W_{read}^{†} W_{read} = I$ . Tracing the orthogonal $E$ -labels removes cross-outcome terms. Apply Theorem thm:global.

Displayed formulas in this passage

unlabeled · line 1341
$W_{read} ∣ ψ ⟩ = x \sum ∣ x ⟩_{K} ∣ x ⟩_{E} M_{x} ∣ ψ ⟩ .$

Nearby in this section

section: The external residual program
theorem: Theorem: Unobserved readout-and-drop is partial trace
section: The external residual program
section: The external residual program
theorem: Theorem: Classical records are not a quantum decoder
section: The external residual program