Entanglement criterion based on skew information

We know that I(̺,M) and uncertainty δ2(M)̺ coincides on pure state, i.e, when ̺ is a pure state, then the inequalities (2) and (6) in lemma 1, 2 both become equality. let us recall the “Local Uncertainty Relations” (LURs), introduced by Hofmann and Takeuchi [4]. Let Ai be observables on Alice’s space of a bipartite system. If they do not share a common eigenstate, there is a number CA > 0 such that ∑ i δ 2(Ai)̺A ≥ CA holds for all states ̺A on Alice’s space. Hofmann and Takeuchi showed: Proposition 1. [4] Let ̺ be separable and let Ai, Bi, i = 1, ..., n be operators on Alice’s (resp. Bob’s) space, fulfilling ∑n i=1 δ 2(Ai)̺A ≥ CA and ∑n i=1 δ 2(Bi)̺B ≥ CB. We define Mi := Ai ⊗ 1+ 1⊗Bi. Then