Closure of quantum permutations under composition

Determine whether the group closure of quantum permutations is the entire group of unitaries whose block matrices have unit row sums and unit column sums.

Background

The paper treats quantum permutations, or magic unitaries, as a class of unitary transformations whose operator-valued blocks are projections and whose rows and columns sum to the identity. It observes that these transformations behave like a quantum group, but their composition properties are more restricted than those of the full unitary group.

The authors identify the characterization of the resulting closure as relevant to determining which transformations can arise from repeated composition of quantum permutations. They state that the closure is contained in the unitaries with unit row and column sums, but leave unresolved whether this containment is an equality; they do establish that it is not the entire unitary group.