bibkey: karlsson2016bccbhadamard
authors: Bengt R. Karlsson
year: 2016
title: "BCCB complex Hadamard matrices of order 9, and MUBs"
doi: 10.1016/j.laa.2016.04.012
url: https://arxiv.org/abs/1604.03770v1
claim: "The paper constructs the two-parameter non-affine orbit BC_9^(2) of symmetric block-circulant complex Hadamard matrices of order 9 with circulant blocks, the locus x+y+u+w=1 of four unimodular parameters, identifies the affine one-parameter suborbits BC_9A^(1) and BC_9B^(1), and conjectures (Conjecture 3) that these are the only affine suborbits."
strata_touched:
  - D5/S3/Quantum/Tomography/MUB/BCCBOrderNineAffineSuborbits
license: citation-only
triage: anchor

BCCB complex Hadamard matrices of order 9, and MUBs

Bengt R. Karlsson, arXiv:1604.03770v1 (quant-ph), 2016-04-13; published in Linear Algebra Appl. 504 (2016) 309–324. Quotations are from the v1 TeX source.

The definition (Section 2):

An orbit of dephased complex Hadamard matrices is affine if the phases of the elements are linear functions of the orbit parameters.

The matrix (Section 4):

\begin{equation} H=\left(\begin{array}{ccccccccc} 1 & x & x & y & u & w & y & w & u\ x & 1 & x & w & y & u & u & y & w\ x & x & 1 & u & w & y & w & u & y\ y & w & u & 1 & x & x & y & u & w\ u & y & w & x & 1 & x & w & y & u\ w & u & y & x & x & 1 & u & w & y\ y & u & w & y & w & u & 1 & x & x\ w & y & u & u & y & w & x & 1 & x\ u & w & y & w & u & y & x & x & 1 \end{array}\right)\label{eq:Basic form} \end{equation}

The unitarity locus (Section 5):

If $\sigma =x+y+u+w=1$, all the unitarity conditions (\ref{unitarity cond}) are satisfied, with no additional constraints on $x,y,u,w\in \mathbb{T}$,

The two families (Sections 8.1 and 8.2):

If $(xyuw)$ is a permutation of $(\mu ,-\mu ,-\omega ,-{\omega }^2),$ with $\mu \in \mathbb{T}$,

If $(xyuw)$ is a permutation of $(1,\xi ,\omega \xi ,{\omega }^2\xi ),$ with $\xi \in \mathbb{T}$,

Conjecture 3 (Section 8.2):

$B{C}_{9A}^{(1)}$ and $BC{}_{9B}^{(1)}$ are the only affine suborbits of $B{C}_9^{(2)}$.

Verified locator

• URL: https://arxiv.org/abs/1604.03770v1 (v1, 2016-04-13; source Hadamard_BC_9__2__final.tex, md5 8e0aaa1643ecbb7f20eb97937e8fa4c8, Latin-1 encoded): the definition (l. 137–138), the matrix (l. 305–317), the unitarity locus (l. 388–389), the two families (l. 551–552, 598–599) and Conjecture 3 (l. 628–631).
• DOI: 10.1016/j.laa.2016.04.012 (Linear Algebra Appl. 504, 309–324). The published text was not read.