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

Proposition: PPT states cannot pass the usual Bell-entanglement test

Finite-sample certification and its limitations › Why a Bell-overlap test is not the required witness · lines 1743–1763 · proof included or linked

Oai272-A@1#prop:pptbell

For a normalized PPT state $σ$ on $C^{d} \otimes C^{d}$ ,

⟨ Φ_{d} ∣ σ ∣ Φ_{d} ⟩ \leq \frac{1}{d} .

Proof. If $S$ is the swap operator, then

(∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣)^{T_{B}} = S / d .

Since $σ^{T_{B}} ⪰ 0$ , $Tr σ^{T_{B}} = 1$ , and $S ⪯ I$ ,

Tr (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣ σ) = Tr [(S / d) σ^{T_{B}}] \leq 1/ d .

Displayed formulas in this passage

unlabeled · line 1747
$⟨ Φ_{d} ∣ σ ∣ Φ_{d} ⟩ \leq \frac{1}{d} .$
unlabeled · line 1754
$(∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣)^{T_{B}} = S / d .$
unlabeled · line 1759
$Tr (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣ σ) = Tr [(S / d) σ^{T_{B}}] \leq 1/ d .$