An algebraic characterization of injectivity in phase retrieval

A complex frame is a collection of vectors that span CMand define measurements, called intensity measurements, onvectors in CM. Inpurely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. Weshow that any vector is uniquely determined (upto a global phase factor) from 4M−4generic measurements. Toprove this, weidentify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.