$BCCB$ complex Hadamard matrices of order 9, and MUBs

TL;DR

This paper introduces a new class of symmetric, block circulant complex Hadamard matrices of order 9, revealing their structure, suborbits, defects, and connections to mutually unbiased bases, expanding the understanding of quantum measurement frameworks.

Contribution

The paper constructs a novel two-parameter family of non-reducible, non-affine complex Hadamard matrices of order 9 with unique properties and links to MUBs.

Findings

Identified a new two-parameter orbit of Hadamard matrices

Discovered suborbits including an intersection with Fourier orbit

Linked certain matrices to complete sets of MUBs in dimension 9

Abstract

A new type of complex Hadamard matrices of order 9 are constructed. The studied matrices are symmetric, block circulant with circulant blocks ($BCCB$) and form an until now unknown non-reducible and non-affine two-parameter orbit. Several suborbits are identified, including a one-parameter intersection with the Fourier orbit $F_{9}^{(4)}$. The defect of this new type of Hadamard matrices is observed to vary, from a generic value 2 to the anomalous values 4 and 10 for some sub-orbits, and to 12 and 16 for some single matrices. The latter matrices are shown to be related to complete sets of MUBs in dimension 9.