Roland Hildebrand

An LMI description for the cone of Lorentz-positive maps

Let Ln be the n-dimensional second-order cone. A linear map from to is called positive if the image of Lm under this map is contained in Ln . For any pair ( n,u2009m ) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality (LMI) that describes this cone. Namely, we show that its dual cone, the cone of Lorentz–Lorentz separable elements, is a section of some cone of positive semidefinite complex hermitian matrices. Therefore the cone of positive maps is a projection of a positive semidefinite matrix cone. The construction of the LMI is based on the spinor representations of the groups . We also show that the positive cone is not hyperbolic for .

Combinatorial laplacians and positivity under partial transpose

The density matrices of graphs are combinatorial laplacians normalised to have trace one (Braunstein et al. 2006b). If the vertices of a graph are arranged as an array, its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for the separability of density matrices of graphs, is equivalent to the PPT-criterion. As such, it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (see Wu (2006) for an alternative proof). Finally, we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.

Scaling relationship between the copositive cone and Parrilo's first level approximation

We investigate the relation between the cone

Irreducible elements of the copositive cone

{mathcal{C}^{n}}

Exactness of sums of squares relaxations involving 3x3 matrices and Lorentz cones

of n × n copositive matrices and the approximating cone

Copositive matrices with circulant zero support set

{mathcal{K}_{n}^{1}}