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

Theorem: Fine-graining anchoring

Diagonal anchoring: independent finite proofs › Proof B: decohered fine-graining · lines 1641–1686 · proof included or linked

I@2#thm:fine-anchor

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Assume $(C)^{⋆}, (NS)^{⋆}, (N)^{⋆}, (K)^{⋆}$ . Then diagonal anchoring holds.

Proof. Use the rational weights above and set

a = ⌈ lo g_{2} M ⌉, b = ⌈ lo g_{2} k max m_{k} ⌉ .

Choose disjoint $R_{k} \subseteq {0, 1}^{a}$ , $∣ R_{k} ∣ = m_{k}$ , and injections

e_{k} : R_{k} ⟶ {0, 1}^{b} .

Define

Φ = \frac{1}{M} k \sum r \in R_{k} \sum ∣ k ⟩_{S} ∣ r ⟩_{E^{'}} ∣ e_{k} (r) ⟩_{E^{''}} .

Its reduction on $S$ is $ρ_{S}$ .

With $E^{'}$ written first, each $r$ has companion $(k (r), e_{k (r)} (r))$ . These companions are distinct: different blocks have different $k$ , and each $e_{k}$ is injective. Uniform correlations therefore give

P^{E^{'}}_{Φ} (r) = 1/ M

on the used labels.

Let $ν = P^{S \cup E^{'}}_{Φ}$ . Nullity restricts its support to $(k (r), r)$ . Composability makes its $E^{'}$ -marginal uniform. Each $r$ has only one allowed companion, so

ν (k (r), r) = 1/ M .

The $S$ -marginal is $m_{k} / M$ . Locality transfers this law to every purification of $ρ_{S}$ , and continuity supplies arbitrary diagonal weights.

Displayed formulas in this passage

unlabeled · line 1649
$a = ⌈ lo g_{2} M ⌉, b = ⌈ lo g_{2} k max m_{k} ⌉ .$
unlabeled · line 1655
$e_{k} : R_{k} ⟶ {0, 1}^{b} .$
unlabeled · line 1659
$Φ = \frac{1}{M} k \sum r \in R_{k} \sum ∣ k ⟩_{S} ∣ r ⟩_{E^{'}} ∣ e_{k} (r) ⟩_{E^{''}} .$
unlabeled · line 1671
$P^{E^{'}}_{Φ} (r) = 1/ M$
unlabeled · line 1680
$ν (k (r), r) = 1/ M .$

Nearby in this section

section: Proof B: decohered fine-graining