  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.
