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 complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last these, for which least is known, in particular understanding polytopes psd-minimal. prove existence an obstruction psd-minimality efficiently computable lattice membership problems. Using this tool, we complete classification psd-minimal polygons (geometrically as well combinatorially). In dimension three exhibit new examples apply our rule out many others.