An LMI description for the cone of Lorentz-positive maps

Abstract

Let L_n be the n-dimensional second-order cone. A linear map from to is called positive if the image of L_m under this map is contained in L_n . For any pair ( n, m ) 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 .

Keywords:

• Second order cone
• Hyperbolic polynomial
• Semidefinite cone

Keywords:

• AMS Subject Classifications: 15A48
• 15A66
• 15A69
• 90C22