Orthogonality of subalgebras and Latin squares

By a ∗-subalgebra of the matrix algebra Mn(C) we mean a subalgebra containing the identity closed under ∗ (conjugate transpose). A maximal abelian ∗-subalgebra of Mn(C) is unitarily conjugate to the subalgebra of diagonal matrices. Two maximal abelian ∗-subalgebras A,B ofMn(C) are said to be orthogonal (or an orthogonal pair) if the following condition is satisfied. tr(x)tr(y) = ntr(xy) for all x ∈ A, y ∈ B. Geometric meaning of the above condition is that A ∩ 1 ⊥ and B ∩ 1 ⊥ are orthogonal with respect to the inner product 〈x|y 〉 = tr(xy∗). For example, the subalgebra A of diagonal matrices is orthogonal to the subalgebra B generated by the matrix