Two Conjectures on Finite Gabor Systems

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Advances in digital signal processing often rely on successes in pure and computational harmonic analysis, a branch of mathematics dating back to Joseph Fourier. Examples of these successes include the introduction of the JPEG standard for compression of photographic images, advances in phaseless reconstruction, and compressed sensing, e.g., the construction of single pixel cameras. At the core of this progress is a better understanding of the redundancy inherent in many data generated in our daily lives. Frame theory can be viewed as one of the appropriate paradigms to investigate and model redundancy. The investigators use frame theory to study two classes of problems whose solutions could have significant applicability in quantum information theory. Because some of the mathematical problems taken up in this project are related to engineering problems, their solutions could lead to advances in signal processing and technological infrastructure, as well as broaden the understanding and role of frames in applications.The investigators study the Zauner conjecture in quantum information theory and the HRT (Heil-Ramanathan-Topiwala) conjecture in time-frequency analysis. They observe that the Zauner conjecture is a special case of the HRT conjecture in the setting of rank-one finite-dimensional time-frequency matrices, and use that relationship initially for the transference of current technology for each conjecture. The theory of frames plays a fundamental role in formulating and understanding the problems the investigators examine here. The notion of the coherence of finite sets of vectors is an important quantitative measure necessary to make technical progress in solving these problems, especially as regards understanding the role of maximal incoherence that such sets may have.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.