# Complex Hadamard matrices with noncommutative entries

## TL;DR

This paper generalizes complex Hadamard matrices to entries in noncommutative $C^*$-algebras, enabling the study of quantum permutation groups and continuous families of such matrices.

## Contribution

It introduces a formalism for complex Hadamard matrices with noncommutative entries and constructs associated quantum permutation groups.

## Key findings

- Established axioms for Hadamard matrices over $C^*$-algebras
- Constructed quantum permutation groups from these matrices
- Connected the formalism to classical and continuous matrix families

## Abstract

We axiomatize and study the matrices of type $H\in M_N(A)$, having unitary entries, $H_{ij}\in U(A)$, and whose rows and columns are subject to orthogonality type conditions. Here $A$ can be any $C^*$-algebra, for instance $A=\mathbb C$, where we obtain the usual complex Hadamard matrices, or $A=C(X)$, where we obtain the continuous families of complex Hadamard matrices. Our formalism allows the construction of a quantum permutation group $G\subset S_N^+$, whose structure and computation is discussed here.