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By R Olkiewicz; et al

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W1 . (19) Clearly, W = W1 Wd . e. states ρ with Schmidt number l < SN(ρ) ≤ k. In particular W ∈ Wk Wk+1 can detect state ρ with SN(ρ) = k. Consider now the following class of witnesses Wrs := Wr + (id ⊗ T)Ws , (20) W = P + QΓ , (21) that is, W ∈ Wrs iff with P ∈ Wr and Q ∈ Ws . Note, that Tr(W ρ) ≥ 0 for all ρ ∈ Vr ∩ Vs . Hence such W can detect PPT states ρ such that SN(ρ) ≥ r or SN(ρΓ ) ≥ s. 27 They cannot detect PPT states. One has the following chain of inclusions: Wdd ⊂ . . ⊂ W22 ⊂ W11 ≡ W .

Clearly, any 2-EW from our class is necessarily decomposable. It was shown 14 that all 2-EW from the class W [a, b, c] are decomposable. 7. Bell-diagonal entanglement witnesses Let us define a generalized Bell states46 in Cd ⊗ Cd ψmn = (Id ⊗ Umn )ψd+ , (66) where Umn are unitary matrices defined as follows Umn ek = λmk ek+n , (67) λ = e2πi/d . (68) with 5th August 2010 52 12:12 WSPC - Proceedings Trim Size: 9in x 6in karp46proc D. Chru´sci´ nski & A. Kossakowski The matrices Umn define an orthonormal basis in the space Md (C) of complex d × d matrices.

It is clear since W [1, 1, 0] is indecomposable. It was shown 14 that W [a, b, c] is decomposable if and only if a ≥ 0 and bc ≥ (2 − a)2 . 4 (65) Hence W [a, b, c] from our spectral class is always decomposable. In particular W [0, 1, 1] reproduces the EW corresponding to the reduction map in d = 3. e. satisfy (65), but do not belong to or spectral class. Similarly one can check when W [a, b, c] defines 2-EW. One finds µ2 = 2(2 − a) and hence condition (46) implies (1) 1 ≤ a < 2, (2) b, c ≥ 2(2 − a).

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