Variational characterizations of separability and entanglement of formation - art. no. 052304

In this paper we develop a mathematical framework for the characterization of separability and entanglement of formation (EOF) of general bipartite st ates. These characterizations are variational in nature, meaning that separ ability and EOF are given in terms of a function that is to be minimized ov er the manifold of unitary matrices. A major benefit of such a characteriza tion is that it directly leads to a numerical procedure for calculating EOF . We present an efficient minimization algorithm and apply it to the bound entangled 3 x 3 Horodecki states: we show that their EOF is very low and th at their distance to the set of separable states is also very small. Within the same variational framework we rephrase the results by Wootters [W. Woo tters. Phys. Rev. Lett. 80, 2245 (1998)] on EOF for 2 x 2 states and also p resent some progress in generalizing these results to higher-dimensional sy stems.