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We present a method to derive separability criteria for the different classes of multiparticle entanglement, especially genuine multiparticle entanglement.
W. Dür and I. Cirac, Phys. Rev. A 61
2000
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The best known witness is 𝒲 = ( 2 / 3 ) ⋅ 𝟙 − | 𝔻 𝟜 ⟩ ⟨ 𝔻 𝟜 | , \mathcal{W}=(2/3)\cdot\openone-|D_{4}\rangle\langle D_{4}|, see G. Tóth, J. Opt. Soc. Am. B 24
2007
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M. Horodecki et al
2008
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For a justification see Section 3.2.2 in Ref. [ 2 ]
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More generally, one has that if f 1 , … , f n f_{1},...,f_{n} are positive concave functions, then g = ( ∏ k = 1 n f k ) 1 / n g=\big(\prod_{k=1}^{n}f_{k}\big)^{1/n} is also concave. This can be seen as follows: First, as the function h ( x ) = ( x ) 1 / n h(x)=(x)^{1/n} is monotonically increasing, it suffices to prove the claim for linear f k . f_{k}. Then, one can directly calculate that the second derivative of g g is not positive. See also p. 87 in S. Boyd and L. Vandenberghe, Convex optimization
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For three qubits, this witness is 𝒲 = ( 2 / 3 ) ⋅ ( 𝟙 − | 𝟙𝟙𝟙 ⟩ ⟨ 𝟙𝟙𝟙 | ) − | 𝕎 𝟛 ⟩ ⟨ 𝕎 𝟛 | , \mathcal{W}=(2/3)\cdot(\openone-|111\rangle\langle 111|)-|W_{3}\rangle\langle W_{3}|, see also Section 6.8.2 in Ref. [ 2 ]
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This generalizes the estimate | ϱ 23 | ≤ ( ϱ 22 + ϱ 33 ) / 2 |\varrho_{23}|\leq(\varrho_{22}+\varrho_{33})/2 from Observation 3 and can be seen as follows: One has for any | x ⟩ |x\rangle that ⟨ x | P | x ⟩ ≥ 0 , \langle x|P|x\rangle\geq 0, and taking | x ⟩ |x\rangle of the type | x ⟩ = ( 1 , e i ϕ , 0 , … 0 ) |x\rangle=(1,e^{i\phi},0,...0) and summing over all possible permutations thereof gives the bound
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Cited in the paper.
M. Seevinck and J. Uffink, Phys. Rev. A 78
2008
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O. Gühne and G. Tóth, Phys. Reports 474
2009
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D. Collins et al
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