We extend the concept of the negativity, a good measure of entanglement for bipartite pure states, to mixed states by means of the convex-roof extension. We show that the measure does not increase under local quantum operations and classical communication, and derive explicit formulae for the entanglement measure of isotropic states and Werner states, applying the formalism presented by Vollbrecht and Werner [Phys. Rev. A 64, 062307 (2001)].

Abstract. We exhibit a two-parameter class of states ρ (α,γ) , in 2 ⊗ n quantum system for n ≥ 3, which can be obtained from an arbitrary state by means of local quantum operations and classical communication, and which are invariant under all bilateral operations on 2 ⊗ n quantum system. We calculate the negativity of ρ (α,γ) , and a lower bound and a tight upper bound on its entanglement of formation. It follows from this calculation that the entanglement of formation of ρ (α,γ) cannot exceed its negativity.

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In this paper, we present sufficient conditions for states to have positive distillable key rate. Exploiting the conditions, we show that the bound entangled states given by Horodecki et al. [Phys. Rev. Lett. 94, 160502 (2005), quant-ph/0506203] have nonzero distillable key rate, and finally exhibit new classes of bound entangled states with positive distillable key rate, but with negative Devetak-Winter lower bound of distillable key rate for the ccq states of their privacy squeezed versions.