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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)].

We derive a class of inequalities, from the uncertainty relations of the su͑1,1͒ and the su͑2͒ algebra in conjunction with partial transposition, that must be satisfied by any separable two-mode states. These inequalities are presented in terms of the su͑2͒ operators J, and the total photon number ͗N a + N b ͘. They include as special cases the inequality derived by Hillery and Zubairy ͓Phys. Rev. Lett. 96, 050503 ͑2006͔͒, and the one by Agarwal and Biswas ͓New J. Phys. 7, 211 ͑2005͔͒. In particular, optimization over the whole inequalities leads to the criterion obtained by Agarwal and Biswas. We show that this optimal criterion can detect entanglement for a broad class of non-Gaussian entangled states, i.e., the su͑2͒ minimum-uncertainty states. Experimental schemes to test the optimal criterion are also discussed, especially the one using linear optical devices and photodetectors.

We investigate entanglement of two electron spins forming Cooper pairs in an s-wave superconductor. The two-electron space-spin density matrix is obtained from the BCS ground state using a two-particle Green's function. It is demonstrated that a two spin state is not given by a spin singlet state but by a Werner state. It is found that the entanglement length, within which two spins are entangled, is not the order of the coherence length but the order of the Fermi wave length.Comment: 4 pages, 2 eps figure