In time-frequency analysis, some of the central objects of study are Gabor frames. A sequence consisting of time-frequency translates of a single function in is called a Gabor system. The Balian-Low theorem, a fundamental theorem about Gabor systems, implies that for sufficiently nice , Gabor systems can never be orthonormal bases. A solution to this is to settle with (Gabor) frames, which can be overcomplete in general.
The theory of Gabor frames is available in for any locally compact abelian group , but has so far been investigated mostly for ``elementary'' groups such as , , , and products of these. As first discovered by F. Luef, Gabor frames in are related to M. Rieffel's Heisenberg modules over twisted group -algebras. In this talk I will show how the Balian-Low theorem, which concerns , might or might not generalize to , and that the answer depends on the structure of certain Heisenberg modules, or equivalently, certain line bundles associated to .
I will also present joint work with Luef and M. Jakobsen on the existence of Gabor frames in , with being either the adeles over the rationals or the group .?
Gabor frames, Heisenberg modules and vector bundles
Nov. 15, 2018 3pm (MATH 350)
Probability
Lu Wei (University of Michigan-Dearborn)
X
Quantum bipartite model is a standard model in describing the interaction of a physical object and its environment for different quantum systems. We wish to understand the degree of entanglement of such systems as measured by von Neumann entropy via its exact moments. The problem of computing the moments of von Neumann entropy was first considered by Don Page in 1994, where a formula for the first moment was conjectured. Page’s conjecture was proved independently by a number of works in the years followed. Recently, an expression for the second moment was conjectured by Vivo, Pato, and Oshanin, which was subsequently proved by us last year. In this talk, we will first review the computations for the first two moments. We will then show how our approach can be extended to obtain the third moment and beyond.
Moments of von Neumann Entropy of Quantum Bipartite Systems: Results in Progress and in Perspectives