Manuscript VII-E@1 · VII-E.tex · theorem · b713cf3c

Theorem: Exact record compatibility

Record compatibility, residuals, and composition › Effect pullback · lines 1125–1142 · proof included or linked

VII-E@1#thm:pullback, VII-E@1#eq:pullback

The two record laws agree for every state if and only if

T^{*} (E_{w}) = Π_{w} (w \in W) .

Proof. Their difference at outcome $w$ is

Tr (ρ [T^{*} (E_{w}) - Π_{w}]) .

The bracket is Hermitian. If its expectation vanishes in every state, testing its eigenvectors shows that every eigenvalue is zero. The reverse implication is the adjoint identity.

Displayed formulas in this passage

VII-E@1#eq:pullback · line 1128
$T^{*} (E_{w}) = Π_{w} (w \in W) .$
unlabeled · line 1136
$Tr (ρ [T^{*} (E_{w}) - Π_{w}]) .$

Nearby in this section

section: Effect pullback
section: Effect pullback