Phase retrieval

Abstract

Abstract. We answer a number of open problems concerning phase retrieval and phase retrieval by projections. In particular, one main theorem classifies phase retrieval by projections via collections of sequences of vectors allowing norm retrieval. Another key result computes the minimal number of vectors needed to add to a frame in order for it to possess the complement property and hence allow phase retrieval. In furthering this idea, in a third main theorem we show that when a collection of subspaces is one subspace short from allowing phase retrieval, then any partition of orthonormal bases from these subspaces into two sets which fail to span, then each spans a hyperplane. We offer many more results in this area as well as provide a large number of examples showing the limitations of the theory.

Norm Retrieval and Phase Retrieval by Projections

“…It is enough to check if the identity is in the linear span of the projections in order for the subspaces to do norm retrieval. A similar result in the case of phase retrieval is proved in [17].…”

“…We use Theorem 9 to give a simple proof of a result in [17] which has a very complicated proof in that paper.…”

Quantum State Tomography of Four-Level Systems with Noisy Measurements

“…The problem of complex vector reconstruction (phase retrieval) appears in many areas of research [1,2]. The goal is to uniquely determine an unknown vector |x ∈ C d based on modulus of inner product of the vector in question with elements of a frame Ξ = {ξ 1 , .…”