An algebraic characterization of injectivity in phase retrieval

A complex frame is a collection of vectors that span C and define measurements, called intensity measurements, on vectors in C . In purely 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. We show that any vector is uniquely determined (up to a global phase factor) from 4M − 4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.