Block-Circulant Complex Hadamard Matrices

TL;DR

This paper introduces a novel block-circulant approach to generate isolated complex Hadamard matrices for dimensions 7 and above, including new matrices of orders 9, 10, and 11, with potential applications in quantum information.

Contribution

It presents a new method based on block-circulant structures for constructing isolated complex Hadamard matrices, including several new examples and multiparametric families for dimensions 9 to 11.

Findings

New isolated matrices of orders 9, 10, and 11 not based on roots of unity.

Several new multiparametric families of order 10.

Novel connections between eight-dimensional matrices and classification insights.

Abstract

A new method of obtaining a sequence of isolated complex Hadamard matrices (CHM) for dimensions $N\geqslant 7$, based on block-circulant structures, is presented. We discuss, several analytic examples resulting from a modification of the Sinkhorn algorithm. In particular, we present new isolated matrices of orders $9$, $10$ and $11$, which elements are not roots of unity, and also several new multiparametric families of order $10$. We note novel connections between certain eight-dimensional matrices and provide new insights towards classification of CHM for $N\geqslant 7$. These contributions can find real applications in Quantum Information Theory and constructions of new families of Mutually Unbiased Bases or Unitary Error Bases.