A universal theory of switching for combinatorial objects, and applications to complex Hadamard matrices
Dean Crnković, Ronan Egan, Andrea Švob
TL;DR
This paper unifies diverse switching notions under a universal framework and extends Orrick’s real-Hadamard switching to Butson and complex Hadamard matrices. It develops a general switching theory with rank-1 and rank-2 constructions, including Fourier-set and generalized Hall-set switchings, enabling new inequivalent BH and CH matrices and illuminating trades. The results connect switching with spectral, combinatorial, and design-theoretic concepts, offering tools for classification and construction of complex Hadamard families and addressing open questions on trade sizes. The work has implications for quantum information (e.g., mutually unbiased bases) and combinatorial matrix theory, providing a versatile methodology for generating and distinguishing inequivalent Hadamard objects.
Abstract
The concept of switching has arisen in several different areas within combinatorics. The act of switching usually transforms a combinatorial object into a non-isomorphic object of the same type, in a way that some key property is preserved. Godsil-McKay switching of graphs preserves the spectrum, switching of designs preserves their parameters, and switching of binary codes preserves the minimum distance. For Hadamard matrices, the switching techniques introduced by Orrick proved to be an incredibly powerful tool when enumerating the Hadamard matrices of order $32$. In this paper, we introduce a universal definition of switching that can be adapted to incorporate these known types of switching. Through this language, we extend Orrick's methods to Butson Hadamard and complex Hadamard matrices. We introduce switchings of these matrices that can be used to construct new, inequivalent matrices. We also consider the concept of trades in complex Hadamard matrices in this terminology, and address an open problem on the permissible size of a trade.