Title: Multi-Unitary Complex Hadamard Matrices Show Chinese title

Title: 多酉复数Hadamard矩阵

Abstract: We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of $2k$ subsystems with $d$ levels each to the set of complex Hadamard matrices of order $N=d^k$. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ($H=H^{\rm R}$ or $H=H^{\rm\Gamma}$), two-unitary ($H^R$ and $H^{\Gamma}$ are unitary), or $k$-unitary. Here $X^{\rm R}$ denotes reshuffling of a matrix $X$ describing a bipartite system, and $X^{\rm \Gamma}$ its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in $1+1$ dimensions. Show Chinese abstract

Abstract: 我们分析具有额外对称约束的实数和复数Hadamard矩阵集合。 特别是，我们将存在具有$2k$个子系统且每个子系统有$d$个层级的极大纠缠多体态的问题与阶数为$N=d^k$的复数Hadamard矩阵集合联系起来。 为此，我们研究这类矩阵的可能子集，这些子集是对偶的、强对偶的（$H=H^{\rm R}$或$H=H^{\rm\Gamma}$），双酉的（$H^R$和$H^{\Gamma}$是酉的），或$k$-酉的。 此处 $X^{\rm R}$ 表示描述双粒子系统的矩阵 $X$ 的重新排列，而 $X^{\rm \Gamma}$ 是其部分转置。 这类矩阵在量子多体理论、张量网络和多粒子量子纠缠分类中具有多种应用，并意味着 $1+1$ 维度中一大类解析可解的量子模型。