Non-separable matrix builders for signal processing, quantum information and MIMO applications
Ted Hurley, Barry Hurley
TL;DR
This work develops systematic, non-separable (entangled) matrix constructions for signal processing and quantum information using two core approaches: COSI-based constructions and Diţă-type matrix tangles. It provides a unifying framework to generate unitary, paraunitary and Hadamard matrices in multidimensional settings, including infinite series, and derives a determinant formula $|T| = |A_1| \, |A_2| \, \cdots \, |A_k| \, |U|^n$ that generalizes the tensor-product case. The methods preserve key properties (unitary, invertible, paraunitary) while enabling new entangled structures and symmetric Hadamard realizations; they also connect to group-ring idempotents and Hadamard constructions via reverse circulant schemes. Applications span paraunitary filter banks, Hadamard design, and full-diversity unitary space-time constellations for MIMO, with infinite design families and potential cryptographic/erasure-resilient benefits.
Abstract
Matrices are built and designed by applying procedures from lower order matrices. Matrix tensor products, direct sums or multiplication of matrices are such procedures and a matrix built from these is said to be a {\em separable} matrix. A {\em non-separable} matrix is a matrix which is not separable and is often referred to as {\em an entangled matrix}. The matrices built may retain properties of the lower order matrices or may also acquire new desired properties not inherent in the constituents. Here design methods for non-separable matrices of required types are derived. These can retain properties of lower order matrices or have new desirable properties. Infinite series of required non-separable matrices are constructible by the general methods. Non-separable matrices are required for applications and other uses; they can capture the structure in a unique way and thus perform much better than separable matrices. General new methods are developed with which to construct {\em multidimensional entangled paraunitary matrices}; these have applications for wavelet and filter bank design. The constructions are in addition used to design new systems of non-separable unitary matrices; these have applications in quantum information theory. Some consequences include the design of full diversity constellations of unitary matrices, which are used in MIMO systems, and methods to design infinite series of special types of Hadamard matrices.