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arXiv:quant-ph/0607073 · Open Systems & Information Dynamics

Constructions of Complex Hadamard Matrices via Tiling Abelian Groups

Máté Matolcsi, Júlia Réffy, Ferenc Szöllősi

Published Sep 1, 2007• 26 citations
arXivHTMLPDF

Abstract

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling of Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent, very general construction of complex Hadamard matrices due to Dita [2] via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabó [8], we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue [10] of complex Hadamard matrices of small order.

DOI
10.1007/s11080-007-9050-6

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Towards a classification of 6×66\times 66×6 complex Hadamard matrices
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In this paper we disprove the Haagerup statement that all complex Hadamard matrices of order five are equivalent with the Fourier matrix F5F_5F5​ by constructing circulant matrices that lead to new Hadamard matrices. An important item is the construction of new mutually unbiased bases that are a basic concept of quantum theory and play an essential role in quantum tomography, quantum cryptografy, teleportation, construction of dense coding schemes, classical signal proccesing, etc.