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

Lemma: Choi separability and entanglement breaking

Choi transfer and product-free range · lines 1120–1142 · proof included or linked

Oai272-A@1#lem:choiEB

A CP map $F$ is entanglement breaking if and only if $J (F)$ is separable. This equivalence does not require trace preservation.

Proof. If $J (F)$ is separable, refine its positive factors spectrally:

J (F) = α \sum ∣ a_{α} ⟩ ⟨ a_{α} ∣ \otimes ∣ b_{α} ⟩ ⟨ b_{α} ∣.

Comparison of matrix-unit blocks gives

F (X) = α \sum Tr [(∣ a_{α} ⟩ ⟨ a_{α} ∣)^{T} X] ∣ b_{α} ⟩ ⟨ b_{α} ∣.

On a positive referenced input this is a sum of positive product operators. Conversely, apply the defining entanglement-breaking property to $∣ Ω_{a} ⟩ ⟨ Ω_{a} ∣$ .

Displayed formulas in this passage

unlabeled · line 1128
$J (F) = α \sum ∣ a_{α} ⟩ ⟨ a_{α} ∣ \otimes ∣ b_{α} ⟩ ⟨ b_{α} ∣.$
unlabeled · line 1134
$F (X) = α \sum Tr [(∣ a_{α} ⟩ ⟨ a_{α} ∣)^{T} X] ∣ b_{α} ⟩ ⟨ b_{α} ∣.$

Lemma: Choi separability and entanglement breaking

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