• Mathematics of Signal Processing
• Workshop on Finite Weyl-Heisenberg Groups

• Schedule
• Participants

The problem of proving, or disproving SIC existence (i.e. maximal sets of equiangular lines in C^d, or symmetric informationally complete positive operator valued measures in the physics literature) in every finite dimension has been the focus of much effort since the work of Zauner in the 1990's, but a solution still eludes us. It is an unusual problem, in that we have a large number of examples which exhibit a wealth of unexpected structure additional to the defining feature of equiangularity. So far we have not been able to use this additional structure to prove existence. However, it is, in its own right, extremely interesting, and it is the subject of this talk. The structure is number theoretic in character. It turns out that SICs, in every case that has been checked, generate a type of number field which plays a central role in algebraic number theory. There is a remarkable interplay between the ordinary geometric symmetries and the number-theoretic (or Galois) symmetries.

It is probably fair to say that if SIC existence is ever proved it would be just as interesting (perhaps even more interesting) to number theorists as it would to physicists and design theorists. The talk will describe this side of the SIC problem. No prior knowledge of Galois theory or number theory will be assumed.

I will provide an overview, aimed at those members of the audience who are unfamiliar with quantum mechanics, of the really quite central role that Weyl-Heisenberg groups play in it. In the spirit of quantum information theory I will concentrate on the finite dimensional cases.

Finite Weyl-Heisenberg groups have long been central to the young and comparatively small field of quantum information theory. Since a few years, the venerable juggernaut of condensed matter physics has also taken notice. This happened in the context of a major new trend: the study of "topological order". Traditionally, phases of extended physical systems have been classified in terms of their symmetries. Ice crystals, e.g., exhibit a discrete translational symmetry that is missing in water. In the past few years, condensed matter theory has identified phases that depend not on local symmetries, but on the topological properties of the space they are defined on. This is surprising, as physical interactions are local, whereas topology describes features that are accessible only through a global lense. Such "topologically ordered" phases are stable with respect to local noise processes and are therefore being investigated as possible "topologically protected" ways of storing quantum information. The paradigmatic example of a topologically ordered system is the "toric code". It is defined in terms of WH operators that are arranged on a surface with non-trivial genus.

In principle, it takes nothing more than basic WH theory to analyze its topological properties. Explaining this is the goal of the talk. As a bonus, we'll see that the toric code features "anyonic" excitations – particles that behave more exotically than bosons or fermions.

Signal design using the structure of the Weyl-Heisenberg group is an important topic in several engineering disciplines. This includes, for example, pulse shaping for robust multicarrier transmission schemes, sequence and code design and radar waveform optimization.

In this talk some signal design and optimization problems relevant for asynchronous wireless communication in doubly-dispersive channels will be discussed. This topic becomes relevant again for supporting adaptive waveforms in the air interface design of the 5th generation of cellular mobile networks.

Pulse shaping with respect to averaged "signal-to-interference and noise ratio" (SINR) for spectrally efficient communication is an interplay between localization and "orthogonality". The localization problem itself can be linked to eigenvalues of localization operators and weighted norms of ambiguity and Wigner functions. The right balance between these two sides and even spectral efficiency is still open.

However, several statements on achievable values of certain time-frequency localization measures and fundamental limits on SINR can be obtained already.

The Weyl-Heisenberg group was born and baptized in the quantum-mechanical setup on the euclidean configuration space in the 1930s. The engineering community came across the WH group in the 1960s where the invention of the fast Fourier transform algorithm led to a naturally discrete setup in terms of finite field extensions most noteably of characteristic 2. However, no trivial, generally valid sampling correspondence to the euclidean setup exists which is particularly unfortunate for the highly symmetrical Wigner-Weyl framework. In this talk i will point out two engineering applications of the discrete WH group of key interest to the automotive community: digitally resonant radar and underspread modelling of induction motors.

The theory of compressed sensing considers the following problem: Let A be an m x n matrix and let x be s-sparse, i.e., all but s of its entries vanish. One seeks to recover x uniquely and efficiently from linear measurements y = Ax, although m is much less than n. A sufficient condition to ensure that this is possible is the Restricted Isometry Property (RIP). A matrix is said to have the RIP, if its restriction to any small subset of the columns acts almost like an isometry. In this talk, we study random Gabor synthesis matrices with respect to the RIP, more precisely, matrices consisting of time-frequency shifts of a random vectors with independent subgaussian entries. This question can be reduced to estimating certain suprema of random variables, which are closely related to suprema of chaos processes. Using generic chaining techniques, we derive a bound for their moments in terms of complexity parameters arising in the theory of empirical processes. As a consequence, we obtain that random Gabor synthesis matrices have the RIP with high probability for embedding dimensions scaling linear in the sparsity up to logarithmic factors. This improves upon previous work by Pfander, Rauhut, and Tropp, who obtained a suboptimal scaling with exponent 3/2.

This is joint work with Shahar Mendelson and Holger Rauhut.

In this talk I would like to indicate the relevance of the finite Heisenberg group for operator algebras and noncommutative geometry. On the one hand as toy model and on the other hand in attempts to approximate continuous models by finite-dimensional ones.

Gabor frame is the set of all time-frequency translates of a complex function and is a fundamental tool in utilizing communications channels with wide applications in time-frequency analysis and signal processing. When the domain of the function is a finite cyclic group of order N, then the Gabor frame forms a design on the complex sphere in N dimensions; when the N² unit vectors that constitute this Gabor frame are pairwise equiangular then the Gabor frame forms a spherical 2-design, and in addition, it has minimal coherence, an ideal property in terms of compressive sensing (whether such an equiangular set exists is also known as the SIC-POVM existence problem, which is open since 1999).

In this talk, we will deal with the question of existence of a Gabor frame such that every N vectors form a basis (the discrete analogue of the HRT conjecture); such a frame is called a full spark Gabor frame. This question was posed by Lawrence, Pfander and Walnut in 2005 and was answered in the affirmative by the speaker in 2013 unconditionally. This result has applications in operator identification, operator sampling, and compressive sensing. Furthermore, we will show that when the domain of the function is an abelian non-cyclic group, then the Gabor frame can never be full spark; in this case it is called spark deficient.

Phase retrieval is a non-convex inverse problem arising in many practical applications, such as diffraction imaging, speech recognition and many more. More precisely, we seek to recover a signal of interest from its intensity measurements with respect to some measurement frame. In practice, phaseless measurements with respect to a Gabor frame are relevant for many applications. We are going to describe the idea of a reconstruction algorithm for the case of Gabor measurement frame and then show how geometric properties of the measurement frame, such as projective uniformity and flatness of the vector of frame coefficients, are related to the robustness of the presented algorithm.

We are going to present some results on flatness of the frame coefficient vector for finite dimensional Gabor frames with random window and then formulate an open problem concerning projective uniformity of a Gabor frame, which is not only important for the robustness of the phase retrieval algorithm, but is of general interest for Gabor analysis.

The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. Such operator bases (frames) are called supersymmetric.

We prove that, in the discrete scenario, an operator basis is supersymmetric iff its symmetry group is a unitary 2-design. Such a highly symmetric operator basis can only appear in odd prime power dimensions besides dimensions 2 and 8. Any such basis is covariant with respect to the Heisenberg-Weyl group.

It suffices to single out a unique discrete Wigner function — the Wootters discrete Wigner function — among all possible quasiprobability representations of quantum mechanics. The three exceptions are tied with three special symmetric informaitonally complete measurements. In the course of our study, we show that the Wootters discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.

Main reference:

• Permutation Symmetry Determines the Discrete Wigner Function
  http://arxiv.org/abs/1504.03773

Additional references:

• Super-symmetric informationally complete measurements
  http://arxiv.org/abs/1412.1099
• Multiqubit Clifford groups are unitary 3-designs
  http://arxiv.org/abs/1510.02619