The quantum algebra of partial Hadamard matrices

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A partial Hadamard matrix is a matrix H∈ Mₘ× ₙ(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→ 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)), in both cases introducing certain criteria for the existence of the suitable completions.