Manuscript Oai272-A@1 · Oai272-A.tex · proposition · a40828a0

Proposition: Separable states illustrate the distinction

Stage 4: hypothesis changes and named-model tests › Product-free range is sufficient, not necessary · lines 1817–1840 · proof included or linked

Oai272-A@1#prop:separableC

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Every finite-dimensional separable density operator belongs to $C$ .

Proof. Write

R = k = 1 \sum^{m} p_{k} ∣ a_{k} ⟩ ⟨ a_{k} ∣ \otimes ∣ b_{k} ⟩ ⟨ b_{k} ∣

with normalized local vectors. Take

χ = k \sum p_{k} ∣ k, k ⟩,

L (X) = k \sum X_{k k} ∣ a_{k} ⟩ ⟨ a_{k} ∣, M (X) = k \sum X_{k k} ∣ b_{k} ⟩ ⟨ b_{k} ∣.

These measure-and-prepare maps are CP and unchanged by input transpose. Their tensor product kills the off-diagonal terms of $∣ χ ⟩ ⟨ χ ∣$ and gives $R$ .

Displayed formulas in this passage

unlabeled · line 1825
$R = k = 1 \sum^{m} p_{k} ∣ a_{k} ⟩ ⟨ a_{k} ∣ \otimes ∣ b_{k} ⟩ ⟨ b_{k} ∣$
unlabeled · line 1829
$χ = k \sum p_{k} ∣ k, k ⟩,$
unlabeled · line 1832
$L (X) = k \sum X_{k k} ∣ a_{k} ⟩ ⟨ a_{k} ∣, M (X) = k \sum X_{k k} ∣ b_{k} ⟩ ⟨ b_{k} ∣.$

Proposition: Separable states illustrate the distinction

Displayed formulas in this passage

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