The goal is to extend Gleason’s notion of a frame function, which essential in his fundamental theorem quantum measurement, more general function acting on 1-tight, so-called, Parseval frames. We refer these functions as Gleason for reason our generalization that positive operator-valued measures (POVMs) are essentially equivalent frames and POVMs arise naturally measurement theory. prove under...

In this dissertation, we study the structure of correlation minimizing frames. A correlation minimizing (N,d)-frame is any uniform Parseval frame of N vectors in dimension, d, such that the largest absolute value of the inner products of any pair of vectors is as small as possible. We call this value the correlation constant. These frames are important as they are optimal for the 2-erasures pro...

for all x∈H . When A = B the frame is said to be tight and if in addition, A = B = 1 it is termed a Parseval frame. When F = { fi}i=1 is a frame, we shall abuse notations and denote by F again, the n×M matrix whose ith column is fi, and where n is the dimension of H . Using this notation, the frame operator is the n×n matrix S=FF∗ where F∗ is the adjoint of F . It is a folklore to note that F i...

Quaternion Fourier transform (QFT) has gained significant attention in recent years due to its effectiveness analyzing multi-dimensional signals and images. This article introduces two-dimensional (2D) right-sided quaternion offset linear canonical (QOLCT), which is the most general form of QFT with additional free parameters. We explore properties 2D QOLCT, including inversion Parseval formula...

Motivated by recent developments in quantum simulation of synthetic dimensions, e.g., optical lattices ultracold atoms, we discuss here d-dimensional periodic, gapped systems for d ≤ 4, with a focus on the topology occupied energy states. We perform this analysis asking whether spectral subspace below gap can be spanned smooth and periodic Bloch functions, corresponding to localized Wannier fun...