Hadamard transform

About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.

A concise guide to complex Hadamard matrices

Abstract: Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices and present a catalogue of inequivalent cases known for dimension N=2,...,16. In particular, we explicitly write down some families of complex Hadamard matrices for N=12,14 and 16, which we could not find in the existing literature.

Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6

Abstract: We study the problem of constructing mutually unbiased bases in dimension 6. This approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions. Our technique suggests the existence of previously unknown symmetries in Karlsson's non-affine family K(2)6 which we confirm analytically. Also, we obtain strong evidence that not more than three mutually unbiased bases can be constructed from pairs which contain members of some non-affine families of complex Hadamard matrices.

Some Results on Matchgates and Holographic Algorithms.

Abstract: We establish a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit [14] and his signature theory of planar-matchgate/matchgrid [16], thus unifying the two theories in expressibility. In [3], we had established a complete characterization of general matchgates, in terms of a set of useful Grassmann-Plucker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planar-matchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we also give a polynomial time algorithm for a simultaneous node-edge deletion problem, using holographic algorithms. Finally we give characterizations of symmetric signatures realizable in the Hadamard basis.

Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps

Abstract: Counterparts of the classical integral and discrete Jensen inequalities and the Hermite-Hadamard inequalities for strongly convex set-valued maps are presented.

Topological degree theory and Caputo–Hadamard fractional boundary value problems

Abstract: We study two hybrid and non-hybrid fractional boundary value problems via the Caputo–Hadamard type derivatives. We seek the existence criteria for these two problems separately. By utilizing the generalized Dhage’s theorem, we derive desired results for an integral structure of solutions for the hybrid problems. Also by considering the special case as a non-hybrid boundary value problem (BVP), we establish other results based on the existing tools in the topological degree theory. In the end of the article, we examine our theoretical results by presenting some numerical examples to show the applicability of the analytical findings.