The phaseless rank of a matrix

TL;DR

This paper introduces the concept of phaseless rank, exploring its mathematical properties, connections to amoebas and semidefinite representations, and applications to complex equiangular lines and polytope complexity.

Contribution

It extends classical results on matrix rank, links phaseless rank to amoebas and convex geometry, and provides new bounds on semidefinite extension complexity.

Findings

Maximal minors form an amoeba basis for their ideal.

New upper bounds on semidefinite extension complexity of polytopes.

Connections established between phaseless rank and equiangular lines.

Abstract

We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this quantity, extending a classic result of Camion and Hoffman and connecting it to the study of amoebas of determinantal varieties and of semidefinite representations of convex sets. As a consequence, we prove that the set of maximal minors of a matrix of indeterminates form an amoeba basis for the ideal they define, and we attain a new upper bound on the complex semidefinite extension complexity of polytopes, dependent only on their number of vertices and facets. We also highlight the connections between the notion of phaseless rank and the problem of finding large sets of complex equiangular lines or mutually unbiased bases.