2006/12/08 by Roland Hildebrand, Hildebrand, Roland
Computer Science · Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Physical sciences #Matrix Theory and Algorithms #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.quant-ph/0612064
openalex publication_date 2006/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ln be the n-dimensional Lorentz cone. A linear map M from Rm to Rn is called Lorentz-positive if M[Lm] is contained in Ln. 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) → H(d) be a positive operator, and let λ1,...,λ4 be the generalized eigenvalues of the pencil σ2(P(X)) - λdet X, in decreasing order, where σ2 is the second symmetric function of the spectrum. Then the concurrence is given by the expression C(P;X) = 2√(σ2(P(X)) - λ2 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 λ2 replaced by the smallest eigenvalue λ4.