Group Frames with Few Distinct Inner Products and Low Coherence

Frame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and data transmission, to spacetime codes for MIMO communications, and to measurement operators in quantum sensing. High-performance codes usually arise from designing frames whose elements have mutually low coherence. Building off the original "group frame" design of Slepian which has since been elaborated in the works of Vale and Waldron, we present several new frame constructions based on cyclic and generalized dihedral groups. Slepian's original construction was based on the premise that group structure allows one to reduce the number of distinct inner pairwise inner products in a frame with $n$ elements from $\frac{n(n-1)}{2}$ to $n-1$. All of our constructions further utilize the group structure to produce tight frames with even fewer distinct inner product values between the frame elements. When $n$ is prime, for example, we use cyclic groups to construct $m$-dimensional frame vectors with at most $\frac{n-1}{m}$ distinct inner products. We use this behavior to bound the coherence of our frames via arguments based on the frame potential, and derive even tighter bounds from combinatorial and algebraic arguments using the group structure alone. In certain cases, we recover well-known Welch bound achieving frames. In cases where the Welch bound has not been achieved, and is not known to be achievable, we obtain frames with close to Welch bound performance

Numerically erasure-robust frames

Given a channel with additive noise and adversarial erasures, the task is to design a frame that allows for stable signal reconstruction from transmitted frame coefficients. To meet these specifications, we introduce numerically erasure-robust frames. We first consider a variety of constructions, including random frames, equiangular tight frames and group frames. Later, we show that arbitrarily large erasure rates necessarily induce numerical instability in signal reconstruction. We conclude with a few observations, including some implications for maximal equiangular tight frames and sparse frames.Comment: 15 page

Constructions of biangular tight frames and their relationships with equiangular tight frames

We study several interesting examples of Biangular Tight Frames (BTFs) - basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) - and examine their relationships with Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one frame angle. We demonstrate a smooth parametrization BTFs, where the corresponding frame angles transform smoothly with the parameter, which "passes through" an ETF answers two questions regarding the rigidity of BTFs. We also develop a general framework of so-called harmonic BTFs and Steiner BTFs - which includes the equiangular cases, surprisingly, the development of this framework leads to a connection with the famous open problem(s) regarding the existence of Mersenne and Fermat primes. Finally, we construct a (chordally) biangular tight set of subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.Comment: 19 page

Low-Coherence Frames from Group Fourier Matrices

Many problems in areas such as compressive sensing and coding theory seek to design a set of equal-norm vectors with large angular separation. This idea is essentially equivalent to constructing a frame with low coherence. The elements of such frames can in turn be used to build high-performance spherical codes, quantum measurement operators, and compressive sensing measurement matrices, to name a few applications. In this work, we allude to the group-frame construction first described by Slepian and further explored in the works of Vale and Waldron. We present a method for selecting representations of a nite group to construct a group frame that achieves low coherence. Our technique produces a tight frame with a small number of distinct inner product values between the frame elements, in a sense approximating a Grassmannian frame. We identify special cases in which our construction yields some previously-known frames with optimal coherence meeting the Welch lower bound, and other cases in which the entries of our frame vectors come from small alphabets. In particular, we apply our technique to the problem choosing a subset of rows of a Hadamard matrix so that the resulting columns form a low-coherence frame. Finally, we give an explicit calculation of the average coherence of our frames, and nd regimes in which they satisfy the Strong Coherence Property described by Mixon, Bajwa, and Calderbank

Higher-dimensional Algebra and Topological Quantum Field Theory

The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a "suspension" operation on n-categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k >= n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n-dimensional unitary extended TQFTs as weak n-functors from the "free stable weak n-category with duals on one object" to the n-category of "n-Hilbert spaces". We conclude by describing n-categorical generalizations of deformation quantization and the quantum double construction.Comment: 36 pages, LaTeX; this version includes all 36 figure

The SIC Question: History and State of Play

Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott's code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.Comment: 16 pages, 1 figure, many references; v3: updating bibliography, dimension eight hundred forty fou