Manuscript VII-F-ST@1 · VII-F-ST.tex · theorem · 9eee7a44

Theorem: Record pullback versus residual compatibility

Observation and recovery separations · lines 1454–1491 · proof included or linked

VII-F-ST@1#thm:pullback

Let $T : B (H) \to B (K)$ be a channel, ${P_{z}}$ a projective resolution, $q$ a coarse-graining, and ${I_{w}}$ an instrument on $K$ . Set

Π_{w} = q (z) = w \sum P_{z}, Δ_{w} (X) = q (z) = w \sum P_{z} X P_{z}, E_{w} = I_{w}^{*} (I) .

The recorded maps

\FF (X) = w \sum ∣ w ⟩ ⟨ w ∣ \otimes T (Δ_{w} (X)), \GG (X) = w \sum ∣ w ⟩ ⟨ w ∣ \otimes I_{w} (T (X))

have identical record laws on every state exactly when

T^{*} (E_{w}) = Π_{w} .

They agree on recorded residual observables $\sum_{w} ∣ w ⟩ ⟨ w ∣ \otimes B_{w}$ , $B_{w} \in O$ , exactly when

q (z) = w \sum P_{z} T^{*} (B) P_{z} = T^{*} (I_{w}^{*} (B)) (w, B \in O) .

The identities persist under untouched reference extension.

Proof. Pair the record probabilities with the input state to obtain $Π_{w}$ and $T^{*} (E_{w})$ . Pair the $w$ -residual block with $B$ to obtain the two displayed adjoints. States separate Hermitian matrices; complex linearity and tensoring give the remaining conclusions.

Displayed formulas in this passage

unlabeled · line 1460
$Π_{w} = q (z) = w \sum P_{z}, Δ_{w} (X) = q (z) = w \sum P_{z} X P_{z}, E_{w} = I_{w}^{*} (I) .$
unlabeled · line 1466
$\FF (X) = w \sum ∣ w ⟩ ⟨ w ∣ \otimes T (Δ_{w} (X)), \GG (X) = w \sum ∣ w ⟩ ⟨ w ∣ \otimes I_{w} (T (X))$
unlabeled · line 1471
$T^{*} (E_{w}) = Π_{w} .$
unlabeled · line 1476
$q (z) = w \sum P_{z} T^{*} (B) P_{z} = T^{*} (I_{w}^{*} (B)) (w, B \in O) .$

Theorem: Record pullback versus residual compatibility

Displayed formulas in this passage

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