Associating vectors in $\CC^n$ with rank 2 projections in $\RR^{2n}$: with applications

Abstract: We will see that vectors in $\CC^n$ have natural analogs as rank 2 projections in $\RR^{2n}$ and that this association transfers many vector properties into properties of rank two projections on $\RR^{2n}$. We believe that this association will answer many open problems in $\CC^n$ where the corresponding problem in $\RR^n$ has already been answered - and vice versa. As a application, we will see that phase retrieval (respectively, phase retrieval by projections) in $\CC^n$ transfers to a variation of phase retrieval by rank 2 projections (respectively, phase retrieval by projections) on $\RR^{2n}$. As a consequence, we will answer the open problem: Give the complex version of Edidin's Theorem \cite{E} which classifies when projections do phase retrieval in $\RR^n$. As another application we answer a longstanding open problem concerning fusion frames by showing that fusion frames in $\CC^n$ associate with fusion frames in $\RR^{2n}$ with twice the dimension. As another application, we will show that a family of mutually unbiased bases in $\CC^n$ has a natural analog as a family of mutually unbiased rank 2 projections in $\RR^{2n}$. 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 $\CC^n$ have an analog as equiangular and biangular rank 2 projections in $\RR^{2n}$.