Parametrizing complex Hadamard matrices

Abstract: The purpose of this paper is to introduce new parametric families of complex Hadamard matrices in two different ways. First, we prove that every real Hadamard matrix of order admits an affine orbit. This settles a recent open problem of Tadej and Życzkowski [W. Tadej, K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006) 133–177], who asked whether a real Hadamard matrix can be isolated among complex ones. In particular, we apply our construction to the only (up to equivalence) real Hadamard matrix of order 12 and show that the arising affine family is different from all previously known examples listed in [W. Tadej, K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006) 133–177]. Second, we recall a well-known construction related to real conference matrices, and show how to introduce an affine parameter in the arising complex Hadamard matrices. This leads to new parametric families of orders 10 and 14. An interesting feature of both of our constructions is that the arising families cannot be obtained via Diţă’s general method [P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (20) (2004) 5355–5374]. Our results extend the recent catalogue of complex Hadamard matrices [W. Tadej, K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006) 133–177], and may lead to direct applications in quantum-information theory. [Copyright &y& Elsevier]