Multi-Unitary Complex Hadamard Matrices

TL;DR

This paper explores special classes of complex Hadamard matrices with symmetry constraints and their connections to entangled quantum states, revealing new structures relevant to quantum information and many-body physics.

Contribution

It introduces the analysis of dual, strongly dual, and k-unitary complex Hadamard matrices and links them to multipartite entanglement and solvable quantum models.

Findings

Identifies subsets of Hadamard matrices with specific symmetry properties.

Establishes connections between these matrices and maximally entangled states.

Highlights applications in quantum many-body theory and tensor networks.

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.