Manuscript En09-A@1 · En09-A.tex · proposition · eafec9de

Proposition: Complete classical coarse-graining

Quantum-computing analogs and selective extensions › Coarse records, failure flags, and local POVMs · lines 928–951 · proof included or linked

En09-A@1#prop:coarse

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Let $q : Z \to Y$ be a declared finite coarse-graining. Reporting only $y = q (k)$ , while retaining $S$ , gives branch maps

I_{y}^{q} (X) = k : q (k) = y \sum D_{k} X D_{k}^{†} .

Their sum is trace preserving. On a positive branch,

ρ_{S} (y) = \frac{\sum _{k : q (k) = y} p _{k} ρ _{S} ( t _{k} )}{\sum _{k : q (k) = y} p _{k}} .

This is generally a mixture of fine conditional states, not the state at a single exact time.

Proof. Classical relabeling adds the corresponding diagonal output blocks. Completeness follows by summing over all $y$ . The normalized formula is division of the resulting subnormalized block by its trace.

Displayed formulas in this passage

unlabeled · line 933
$I_{y}^{q} (X) = k : q (k) = y \sum D_{k} X D_{k}^{†} .$
unlabeled · line 937
$ρ_{S} (y) = \frac{\sum _{k : q (k) = y} p _{k} ρ _{S} ( t _{k} )}{\sum _{k : q (k) = y} p _{k}} .$

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section: Coarse records, failure flags, and local POVMs