Phase retrieval by vectors and projections

Abstract

We will see that vectors in C n have natural analogs as rank 2 projections in R 2n and that this association transfers many vector properties into properties of rank two projections on R 2n . We believe that this association will answer many open problems in C n where the corresponding problem in R n has already been answered -and vice versa. As a application, we will see that phase retrieval (respectively, phase retrieval by projections) in C n transfers to a variation of phase retrieval by rank 2 projections (respectively, phase retrieval by projections) on R 2n . As a consequence, we will answer the open problem: Give the complex version of Edidin's Theorem [12] which classifies when projections do phase retrieval in R n . As another application we answer a longstanding open problem concerning fusion frames by showing that fusion frames in C n associate with fusion frames in R 2n with twice the dimension. As another application, we will show that a family of mutually unbiased bases in C n has a natural analog as a family of mutually unbiased rank 2 projections in R 2n . The importance here is that there are very few real mutually unbiased bases but now there are unlimited numbers of real mutually unbiased rank 2 projections to be used in their place. As another application, we will give a variaton of Edidin's theorem which gives a surprising classification of norm retrieval. Finally, we will show that equiangular and biangular frames in C n have an analog as equiangular and biangular rank 2 projections in R 2n .

Phase retrieval and norm retrieval

“…Recovering the phase of a signal through the use of a redundant system of vectors (i.e. a frame) or a spanning collection of subspaces is currently a well studied topic (see [9] for a survey of this field).…”

“…We will also give a number of examples throughout showing that these results are best possible. For an up to date review of phase retrieval by vectors and projections please see [9].…”

Norm Retrievable Frames in $\mathbb{R}^n$

“…The mathematical study of phase retrievable frames was started in 2006 in a landmark paper of Balan, Casazza and Edidin [4]. Also in [9], the authors highlighted recently the major advances in phase retrievable frames.…”

Quantum Tomography of Pure States with Projective Measurements Distorted by Experimental Noise

“…Thus, there is vast literature concerning phase re-trieval, see a review article Ref. [14]. In particular, in recent years, a lot of attention has been paid to the connection between complex vector reconstruction and the theory of frames, e.g.…”