The quantum algebra of partial Hadamard matrices

Abstract

A partial Hadamard matrix is a matrix $H\in M_{M\times N}(\mathbb{T})$ whose rows are pairwise orthogonal. We associate to each such $H$ a certain quantum semigroup $G$ of quantum partial permutations of $\{1,...,M\}$ and study the correspondence $H\to G$. We discuss as well the relation between the completion problems for a given partial Hadamard matrix and completion problems for the associated submagic matrix $P\in M_M(M_N(\mathbb{C}))$, in both cases introducing certain criteria for the existence of the suitable completions.

15 pages; v2 adds more details of the arguments and updates the references. The paper will appear in the Linear Algebra and its Applications