Concurrence of Lorentz-positive maps

Let H(d) be the space of complex hermitian matrices of size d×d and let H+(d) ⊂ H(d) be the cone of positive semidefinite matrices. A linear operator Φ : H(d1) → H(d2) is said to be positive if Φ[H+(d1)] ⊂ H+(d2). The concurrence C(Φ; ·) of a positive operator Φ : H(d1) → H(d2) is a real-valued function on the cone H+(d1), defined as the largest convex function which coincides with 2 q σ d2 2 (Φ(ξξ )) on all rank 1 matrices ξξ∗ ∈ H+(d1). Here σ d 2 : H(d) → R denotes the second symmetric function, defined by σ 2(A) = P i<j μiμj , where μ1, . . . , μd are the eigenvalues of A. The concurrence of a bipartite density matrix X is defined as the concurrence C(Φ;X) with Φ being the partial trace. A analogous concept can be considered for Lorentz-positive maps. Let Ln ⊂ R n be the ndimensional Lorentz cone. Then a linear map Υ : R → R is called Lorentz-positive if Υ[Lm] ⊂ Ln. For this class of maps we are able to compute the concurrence explicitly. This allows us to obtain formulae for the concurrence of positive operators having H(2) as input space and consequently of bipartite density matrices of rank 2. Namely, let Φ : H(2) → H(d2) be a positive operator, and let λ1, . . . , λ4 be the generalized eigenvalues of the pencil σ d2 2 (Φ(X)) − λ detX, in decreasing order. Then the concurrence is given by the expression C(Φ;X) = 2 q σ d2 2 (Φ(X))− λ2 detX . 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 λ2 replaced by the smallest generalized eigenvalue λ4.