Manuscript I@2 · I.tex · section · 85631fa7

Why naive fine-graining does not suffice

Diagonal anchoring: independent finite proofs › Why naive fine-graining does not suffice · lines 1616–1638

For disjoint label sets $R_{k}$ , $∣ R_{k} ∣ = m_{k}$ , the vector

Φ_{0} = \frac{1}{M} k \sum r \in R_{k} \sum ∣ k ⟩_{S} ∣ r ⟩_{E^{'}}

has the desired reduction on $S$ . However,

ρ_{E^{'}} = \frac{1}{M} k \sum r, r^{'} \in R_{k} \sum ∣ r ⟩ ⟨ r^{'} ∣ .

Coherences remain within each block. Distinct refined labels $r$ have repeated companions $k (r)$ ; Theorem thm:uniformity therefore does not apply to the $E^{'}$ -readout.

A further computational label distinguishes those companions. This chooses another purification of the same $ρ_{S}$ ; it does not claim that a remote unitary decoheres a fixed reduced state.

Displayed formulas in this passage

unlabeled · line 1619
$Φ_{0} = \frac{1}{M} k \sum r \in R_{k} \sum ∣ k ⟩_{S} ∣ r ⟩_{E^{'}}$
unlabeled · line 1625
$ρ_{E^{'}} = \frac{1}{M} k \sum r, r^{'} \in R_{k} \sum ∣ r ⟩ ⟨ r^{'} ∣ .$