Geometrical aspects of entanglement

X iv :q ua nt -p h/ 06 05 07 9 v1 8 M ay 2 00 6 Geometrical aspects of entanglement Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum (a) Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway (b) Centre of Mathematics for Applications, P.O. Box 1053 Blindern, 0316 Oslo, Norway and (c) Department of Physics, The Norwegian University of Science and Technology, 7491 Trondheim, Norway (Dated: May 8, 2006) Abstract We study geometrical aspects of entanglement, with the Hilbert–Schmidt norm defining the metric on the set of density matrices. We focus first on the simplest case of two two-level systems and show that a “relativistic” formulation leads to a complete analysis of the question of separability. Our approach is based on Schmidt decomposition of density matrices for a composite system and non-unitary transformations to a standard form. The positivity of the density matrices is crucial for the method to work. A similar approach works to some extent in higher dimensions, but is a less powerful tool. We further present a numerical method for examining separability, and illustrate the method by a numerical study of bound entanglement in a composite system of two three-level systems.