Multi-Unitary Complex Hadamard Matrices

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 = dk. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual (H = HR or H = H Gamma), two-unitary (HR and H Gamma are unitary), or k-unitary. Here XR denotes reshuffling of a matrix X describing a bipartite system, and X 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.