Concurrence of Lorentz-positive maps

Abstract

Let L_n be the n-dimensional Lorentz cone. A linear map M from R^m to R^n is called Lorentz-positive if M[L_m] is contained in L_n. We extend the notion of concurrence, which was initially introduced to quantify the entanglement of bipartite density matrices, to Lorentz-positive maps and provide an explicite formula for it. This allows us to obtain formulae for the concurrence of arbitrary positive operators taking 2 x 2 complex hermitian matrices as input and consequently of arbitrary bipartite density matrices of rank 2. Namely, let P: H(2) \to H(d) be a positive operator, and let λ_1,...,λ_4 be the generalized eigenvalues of the pencil σ_2(P(X)) - λdet X, in decreasing order, where σ_2 is the second symmetric function of the spectrum. Then the concurrence is given by the expression C(P;X) = 2\sqrt{σ_2(P(X)) - λ_2 det X}. As an application, we compute the concurrences of the density matrices of all graphs with 2 edges. Similar results apply for a function which we call I-fidelity, with the second largest generalized eigenvalue λ_2 replaced by the smallest eigenvalue λ_4.