In this paper we develop a mathematical framework for the characterization of separability and entanglement of formation (EOF) of general bipartite states. These characterizations are variational in nature, meaning that separability and EOF are given in terms of a function that is to be minimized over the manifold of unitary matrices. A major benefit of such a characterization 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 $3\times 3$ Horodecki states; we show that their EOF is very low and that their distance to the set of separable states is also very small. Within the same variational framework we rephrase the results by Wootters [W. Wootters, Phys. Rev. Lett. 80, 2245 (1998)] on EOF for $2\times 2$ states and also present some progress in generalizing these results to higher-dimensional systems.

• Received 20 February 2001

DOI:https://doi.org/10.1103/PhysRevA.64.052304