A quantum Latin square whose rows and columns are indexed by finite groups G and G' and whose entry-to-entry inner products are invariant under the group actions, so that the whole array is determined by a single block of data. The central result identifies these invariant quantum Latin squares, up to a global isometry, with trace- and conjugate-transpose-preserving isomorphisms of the group algebras of G and G', giving an existence criterion in terms of equal multisets of irreducible-representation degrees and a bridge to quantum isomorphism of Cayley graphs.
A quantum Latin square is an n-by-n array of unit vectors in a fixed complex vector space in which each row and each column forms an orthonormal basis, generalising an ordinary Latin square. This pap…