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.
Categories: quant-ph math-ph math.MP