Multi-Unitary Complex Hadamard Matrices

• URL: http://arxiv.org/abs/2306.00999v2
• Date: Fri, 14 Jun 2024 18:45:37 GMT
• Title: Multi-Unitary Complex Hadamard Matrices
• Authors: Wojciech Bruzda, Grzegorz Rajchel-Mieldzioć, Karol Życzkowski,
• Abstract summary: We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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