34,631 research outputs found
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
elements from to . 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 is prime, for
example, we use cyclic groups to construct -dimensional frame vectors with
at most 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,
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