The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists in engineering: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. There exist semidefinite programming algorithms to solve this problem, but much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps toward these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with 2M − 1 projections of arbitrary rank in H M . Finally we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.

Phase retrieval has become a very active area of research. We will classify when phase retrieval by Parseval frames passes to the Naimark complement and when phase retrieval by projections passes to the orthogonal complements. We introduce a new concept we call norm retrieval and show that this is what is necessary for passing phase retrieval to complements. This leads to a detailed study of norm retrieval and its relationship to phase retrieval. One fundamental result: a frame {ϕ i } M i=1 yields phase retrieval if and only if {T ϕ i } M i=1 yields norm retrieval for every invertible operator T .1991 Mathematics Subject Classification. Primary 32C15 .

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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 .

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.

Abstract. In a landmark paper [9], Casazza, Fickus, Mixon, Wang and Zhou introduced a fundamental method for constructing unit norm tight frames, which they called Spectral Tetris. This was a significant advancement for finite frame theory -especially constructions of finite frames. This paper then generated a vast amount of literature as Spectral Tetris was steadily developed, refined, and generalized until today we have a complete picture of what are the broad applications as well as the limitations of Spectral Tetris. In this paper, we will put this vast body of literature into a coherent theory.

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