Quantum magic squares were recently introduced as a ‘magical’ combination of
quantum measurements. In contrast to quantum measurements, they cannot be
purified (i.e. dilated to a quantum permutation matrix) — only the so-called
semiclassical ones can. Purifying establishes a relation to an ideal world of
fundamental theoretical and practical importance; the opposite of purifying is
described by the matrix convex hull. In this work, we prove that semiclassical
magic squares can be purified to quantum Latin squares, which are ‘magical’
combinations of orthonormal bases. Conversely, we prove that the matrix convex
hull of quantum Latin squares is larger than the semiclassical ones. This
tension is resolved by our third result: We prove that the quantum Latin
squares that are semiclassical are precisely those constructed from a classical
Latin square. Our work sheds light on the internal structure of quantum magic
squares, on how this is affected by the matrix convex hull, and, more
generally, on the nature of the ‘magical’ composition rule, both at the
semiclassical and quantum level.

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