Magic squares: Latin, semiclassical, and quantum.

The article focuses on the mathematical study of quantum magic squares—matrices whose entries are positive semidefinite operators summing to the identity along each row and column—and their relation to quantum Latin squares, which are matrices of unit vectors forming orthonormal bases in each row and column. It establishes that semiclassical magic squares, defined as convex combinations of permutation matrices tensored with positive operator valued measures (POVMs), can be purified to quantum Latin squares, and that the matrix convex hull of quantum Latin squares is strictly larger than that of semiclassical ones, due to the existence of non-semiclassical quantum Latin squares for sizes four and above. Furthermore, the work shows that quantum Latin squares arising directly from classical Latin squares correspond exactly to the semiclassical magic squares, thereby clarifying the structural distinctions between these classes within the framework of free convexity and operator algebra.