Variational Characterisations of Separability and Entanglement of Formation

In this paper we develop a mathematical framework for the characterisation of separability and entanglement of formation (EoF) of general bipartite states. These characterisations are of the variational kind, meaning that separability and EoF are given in terms of a function which is to be minimized over the manifold of unitary matrices. A major benefit of such a characterisation is that it directly leads to a numerical procedure for calculating EoF. We present an efficient minimisation algorithm and an apply it to the bound entangled 3X3 Horodecki states; we show that their EoF is very low and that their distance to the set of separable states is also very low. Within the same variational framework we rephrase the results by Wootters (W. Wootters, Phys. Rev. Lett. 80, 2245 (1998)) on EoF for 2X2 states and present progress in generalising these results to higher dimensional systems.