Manuscript VII-E-ST@1 · VII-E-ST.tex · lemma · d28d9c5d

Lemma: Sharp-sector intertwining

Contract E: readout, records, and residuals › The exact sharp-measurement criterion · lines 508–530 · proof included or linked

VII-E-ST@1#lem:sharp

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Under (eq:record-pullback), every Kraus operator $A_{ν}$ of $T$ satisfies

E_{w} A_{ν} = A_{ν} Π_{w} .

Consequently

E_{w} T (X) E_{v} = T (Π_{w} X Π_{v}) .

Proof. For $ψ \in im Π_{w}$ ,

ν \sum ∥ (I - E_{w}) A_{ν} ψ ∥^{2} = ⟨ ψ, T^{*} (I - E_{w}) ψ ⟩ = 0.

Thus $A_{ν} Π_{w} = E_{w} A_{ν} Π_{w}$ . For $ψ \in ker Π_{w}$ , the same calculation with $E_{w}$ gives $E_{w} A_{ν} ψ = 0$ . Combine the two identities.

Displayed formulas in this passage

unlabeled · line 512
$E_{w} A_{ν} = A_{ν} Π_{w} .$
unlabeled · line 516
$E_{w} T (X) E_{v} = T (Π_{w} X Π_{v}) .$
unlabeled · line 523
$ν \sum ∥ (I - E_{w}) A_{ν} ψ ∥^{2} = ⟨ ψ, T^{*} (I - E_{w}) ψ ⟩ = 0.$

Nearby in this section

section: The exact sharp-measurement criterion
theorem: Theorem: Lüders compatibility
section: The exact sharp-measurement criterion