关于$6\times 6$复Hadamard矩阵的分类
Towards a classification of $6\times 6$ complex Hadamard matrices
摘要：
复Hadamard矩阵由于在量子信息理论中的出现，在过去的几年里受到了相当大的关注。虽然完整的表征目前只有5阶（\cite{haagerup}），但最近\cite{dita, karol, BN, MM, sz}出现了几种高阶矩阵的新构造。特别地，Beuachamp和Nicoara在\cite{BN}中完成了6阶{ \it自伴}复Hadamard矩阵的分类，提供了以前未知的非仿射单参数轨道。在本文中，我们分类所有\emph{dephased, symmetric}复Hadamard矩阵\emph{with real diagonal}的阶6。进一步放宽对角元的条件，得到了连接Fourier矩阵$F_6$和Different矩阵$D_6$的一个新的非仿射单参数轨道.这回答了Bengtsson等人最近在\cite{BB}中提出的一个问题。
英文摘要：
Complex Hadamard matrices have received considerable attention in the past few years due to their appearance in quantum information theory. While a complete characterization is currently available only up to order 5 (in \cite{haagerup}), several new constructions of higher order matrices have appeared recently \cite{dita, karol, BN, MM, sz}. In particular, the classification of {\it self-adjoint} complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in \cite{BN}, providing a previously unknown non-affine one-parameter orbit. In this paper we classify all \emph{dephased, symmetric} complex Hadamard matrices \emph{with real diagonal} of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix $F_6$ and Diţă's matrix $D_6$. This answers a recent question of Bengtsson & al. in \cite{BB}.
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