Localizable Entanglement and Partial K-way Negativities of Four Qubit States

We use selective partial tansposition to construct partial K- way negativities (K is 2 to 4) that measure the bi-partite, tripartite, and genuine 4-partite entanglement of single copy four qubit states in normal form. For a state in normal form, the partial K-way negativities are polynomial functions of local invariants characterizing the state, as such proper entanglement measures. Nine families of four qubit states, obtained by Versraete et al. [F. Versraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A65, 052112 (2002)], are grouped in two distinct classes that is, (i) states with zero three-way partial negativity and, (ii) states with finite three-way partial negativity. We derive relations between the contribution of a K-way partial transpose to negativity of global partial transpose and the optimum localizable entanglement that may be filtered out from the state through qubit state measurement and classical communication.