Factorization through Lorentz cones

Abstract

A pair of proper cones $({\mathsf{C}}_1,{\mathsf{C}}_2)$ is said to have the Lorentz factorization property (LFP) if every $({\mathsf{C}}_1,{\mathsf{C}}_2)$-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, $({\mathsf{C}}_1,{\mathsf{C}}_2)$ has the LFP if either ${\mathsf{C}}_1$ or ${\mathsf{C}}_2$ 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 $({\mathsf{C}}_1,{\mathsf{C}}_2)$ where ${\mathsf{C}}_1={\mathsf{C}}_2$, or in the case where both ${\mathsf{C}}_1$ and ${\mathsf{C}}_2$ are polyhedral. We also focus on the case where ${\mathsf{C}}_1={\mathsf{C}}_{\square }$ is the square-based cone in ${\mathbf{R}}^3$. Here, we show that $({\mathsf{C}}_{\square },\mathsf{C})$ has the LFP whenever $\mathsf{C}$ 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 $3\times 3$ 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 $\mathsf{C}$ or for a cone $\mathsf{C}$ with $\text{dim}(\mathsf{C})\le 5$. Finally, we discuss an application to a problem in quantum information theory.

21 pages. Comments are welcome