The quantum algebra of partial Hadamard matrices

A partial Hadamard matrix is a matrix H∈MM×N(T)H∈MM×N(T) whose rows are pairwise orthogonal. We associate with each such H a certain quantum semigroup G   of quantum partial permutations of {1,…,M}{1,…,M} and study the correspondence H→GH→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∈MM(MN(C))P∈MM(MN(C)), in both cases introducing certain criteria for the existence of the suitable completions.