Complex Hadamard matrices with noncommutative entries

Abstract

We axiomatize and study the matrices of type $H\in M_N\left(A\right)$ having unitary entries $H_{ij}\in U\left(A\right)$ and whose rows and columns are subject to orthogonality-type conditions. Here $A$ can be any $C^{\ast }$-algebra, for instance $A=\mathbb{C}$, where we obtain the usual complex Hadamard matrices, or $A=C\left(X\right)$, 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 are discussed here.