Letter to the EditorAn algebraic characterization of injectivity in phase retrieval

- We describe polynomial equations that characterize injective intensity measurements.
- We prove that 4M−4 generic intensity measurements suffice for phase retrieval in CM.
- When M=2k+1, 4M−4 measurements are necessary for phase retrieval.

A complex frame is a collection of vectors that span CM and define measurements, called intensity measurements, on vectors in CM. 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.