Speaker
Alexander Muller-Hermes, University of Oslo
Abstract
A pair of proper cones C_1,C_2 is said to have the Lorentz factorization property (LFP) if every (C_1,C_2)-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, (C_1,C_2) has the LFP if either C_1 or C_2 is a direct sum of Lorentzian cones, and our main goal is to find other examples. We show that such examples cannot be found for pairs (C_1,C_2) where C_1=C_2, or in the case where both C_1 and C_2 are polyhedral. Focusing on the case where C_1=C_square is the square-based cone in R^3. Here, we show that (C_square,C) has the LFP whenever C is a symmetric cone, i.e., a direct sum of Lorentz cones, cones of positive semidefinite matrices over the real numbers, complex numbers or quaternions, and the cone of 3-by-3 positive semidefinite matrices over the octonions. We leave open the question whether there are more examples, but we show that this list cannot be extended by any strictly convex cone C or for cones with dim(C)<= 5. Finally, we discuss an application to a problem in quantum information theory. Joint work with Guillaume Aubrun and Francesca La Piana.