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

Lemma: Reduced-state continuity

Locality and uniform correlations · lines 1443–1459 · proof included or linked

I@2#lem:reduced-continuity

Under $(NS)^{⋆}, (K)^{⋆}$ , $F_{S}$ is continuous on the density operators on $H_{S}$ .

Proof. Use the fixed-dimensional purification

∣ ψ_{ρ} ⟩ = (ρ \otimes I) j \sum ∣ j ⟩ ∣ j ⟩ .

Its norm is one and its reduction is $ρ$ . Matrix square root is continuous in finite dimension, so this is a continuous choice of vector purification. Apply $(K)^{⋆}$ and purification locality.

Displayed formulas in this passage

unlabeled · line 1451
$∣ ψ_{ρ} ⟩ = (ρ \otimes I) j \sum ∣ j ⟩ ∣ j ⟩ .$

Nearby in this section

section: Locality and uniform correlations
lemma: Lemma: Remote invariance
lemma: Lemma: Purification locality
section: Locality and uniform correlations
theorem: Theorem: Uniform correlations
section: Locality and uniform correlations