Quantum Latin squares and unitary error bases

We introduce quantum Latin squares, combinatorial quantum objects which generalize classical Latin squares. We show that quantum Latin squares can be seen as weakened versions of mutually-unbiased bases (MUBs). Our main results use quantum Latin squares to give a new construction of unitary error bases (UEBs), basic structures in quantum information which lie at the heart of procedures such as teleportation, dense coding and error correction.