Embedding and Extension Properties of Hadamard Matrices Revisited

Abstract

Hadamard matrices have many applications in several mathematical areas due to their special form and the numerous properties that characterize them. Based on a recently developed relation between minors of Hadamard matrices and using tools from calculus and elementary number theory, this work highlights a direct way to investigate the conditions under which an Hadamard matrix of order n − k can or cannot be embedded in an Hadamard matrix of order n. The results obtained also provide answers to the problem of determining the values of the spectrum of the determinant function for specific orders of minors of Hadamard matrices by introducing an analytic formula.

Exploiting finite geometries for better quantum advantages in Mermin-like games

“…Here I 3 (H ) = Det(A), with A a 3 × 3 matrix with entries ±1. As is known [32] about such 'Hadamard' matrices, the determinant is valued at either 0 or 4. The upper bound for the NCHV model in (21) gives 6 − 2 • 1 = 4 as expected.…”