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

Lemma: Purification locality

Locality and uniform correlations · lines 1408–1432 · proof included or linked

I@2#lem:locality

Under $(NS)^{⋆}$ , global pure states with the same reduced density operator on $S$ have the same $S$ -outcome law.

Proof. Pad the smaller environment by zero-state qubits until environment dimensions agree. Remote invariance leaves the law unchanged; consistent relabeling identifies the environment name sets. If

ρ_{S} = j \sum r_{j} ∣ e_{j} ⟩ ⟨ e_{j} ∣

on its support, the purifications have the forms

j \sum r_{j} ∣ e_{j} ⟩ ∣ f_{j} ⟩, j \sum r_{j} ∣ e_{j} ⟩ ∣ g_{j} ⟩,

with orthonormal environment families. The isometry $f_{j} \mapsto g_{j}$ extends to an environment unitary. Apply remote invariance again.

Displayed formulas in this passage

unlabeled · line 1420
$ρ_{S} = j \sum r_{j} ∣ e_{j} ⟩ ⟨ e_{j} ∣$
unlabeled · line 1424
$j \sum r_{j} ∣ e_{j} ⟩ ∣ f_{j} ⟩, j \sum r_{j} ∣ e_{j} ⟩ ∣ g_{j} ⟩,$

Lemma: Purification locality

Displayed formulas in this passage

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