Complex psd-minimal polytopes in dimensions two and three

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last of these, for which the least is known, and in particular on understanding which polytopes are complex psd-minimal. We prove the existence of an obstruction to complex psd-minimality which is efficiently computable via lattice membership problems. Using this tool, we complete the classification of complex psd-minimal polygons (geometrically as well as combinatorially). In dimension three we exhibit several new examples of complex psd-minimal polytopes and apply our obstruction to rule out many others.

Automated summary

The research focuses on the characteristics of complex psd-minimal polytopes. An efficiently computable obstruction to complex psd-minimality is proved to exist, and this tool is used to complete the classification of complex psd-minimal polygons. Several new examples of complex psd-minimal polytopes are presented in three-dimensional space, and our obstruction is applied to rule out many other cases.