arXiv:1210.7673 · Journal of Mathematical Physics

A new method to construct families of complex Hadamard matrices in even dimensions

Dardo Goyeneche

Published Mar 1, 2013• 11 citations

Abstract

We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with Diţă’s construction and generalizes Szöllősi's method. We extend some known families and present new ones existing in even dimensions. In particular, we find more than 13 millon inequivalent affine families in dimension 32. We also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six and for any family of complex Hadamard matrices existing in every odd dimension.

DOI: 10.1063/1.4794068

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arXiv:1210.7673 · Journal of Mathematical Physics

A new method to construct families of complex Hadamard matrices in even dimensions

Dardo Goyeneche

Published Mar 1, 2013• 11 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/1.4794068

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We show that

K_6^{(2)} \cap X_6^{(2)} = D_6^{(1)} \cup C_6^{(1)}

up to standard Hadamard equivalence, where

C_6^{(1)}

is a new non-affine one-parameter family of complex Hadamard matrices. This demonstrates that the intersection of the two-parameter Hadamard families of Karlsson and Szöllősi is strictly larger than the known affine Diţă curve

D_6^{(1)}

. Our new parametrisation of the Szöllősi family

X_6^{(2)}

in Karlsson coordinates and our constructive classification of the Karlsson-Szöllősi intersection shed light on the intersection of block-reducible forms and hypocycloid constraints. This may support the search for mutually unbiased bases in six dimensions.

3 \times 3

corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.