Concurrence revisited
Résumé
Concurrence is a widely used entanglement measure of bipartite mixed states. We propose to consider the concurrence as being defined by a quadratic form on the space of self-adjoint operators. The square root of this form determines the values of the concurrence on the pure states, while the values on the mixed states are obtained as the largest convex extension, the convex roof. This viewpoint admits a generalization. Namely, the space of self-adjoint operators and the convex cone of positive semidefinite operators contained therein can be replaced by an arbitrary real vector space containing a convex cone. Then the concurrence is determined on the extreme rays of the cone by the square root of a quadratic form, and on the rest of the cone by the convex roof. We compute this generalized concurrence in the case when the cone is a second order cone. This enables us to compute the concurrence of arbitrary bipartite mixed states of rank 2. As an application, we compute the concurrences of the density matrices of all graphs with two edges or with three edges forming a triangle. We also consider the problem of maximizing the concurrence on the set of mixed states having a fixed spectrum.