Low rank matrix recovery from few orthonormal basis measurements

Recent insights concerning the PhaseLift algorithm for retrieving phases have furthered our understanding of low rank matrix recovery from rank-one projective measurements. Motivated by the structure of certain quantum mechanical experiments, we introduce a particular class of such rank-one measurements: orthonormal basis measurements. One such measurement corresponds to choosing an orthonormal basis and treating all the rank-one projectors onto different basis elements as a series of consecutive measurement matrices. We elaborate on performing low-rank matrix recovery from few, sufficiently random orthonormal basis measurements and sketch applications of such a procedure in quantum physics. We conclude this article by presenting numerical experiments testing such an approach.