Complex psd-minimal polytopes in dimensions two and three

TL;DR

This paper investigates complex psd-minimal polytopes, providing a classification for polygons and new examples in three dimensions, along with an obstruction method to determine minimality efficiently.

Contribution

It introduces an obstruction for complex psd-minimality, completes the classification of polygons, and identifies new minimal polytopes in three dimensions.

Findings

Complete classification of complex psd-minimal polygons.

Existence of an efficiently computable obstruction for minimality.

Discovery of new complex psd-minimal polytopes in three dimensions.

Abstract

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.