Triplets of mutually unbiased bases

We initiate a systematic study of triplets of mutually unbiased bases (MUBs). We show that each MUB-triplet in Cᵈ C d is characterized by a d× d× d d × d × d object that we call a Hadamard cube . We describe the basic properties of Hadamard cubes and show how a MUB-triplet can be reconstructed from such a cube, up to unitary equivalence. We also present an algebraic identity which is conjectured to hold for all MUB-triplets in dimension 6. If true, it would imply the long-standing conjecture of Zauner that the maximum number of MUBs in dimension 6 is three.