We study sets of pure states in a Hilbert space of dimension $d$ which are mutually unbiased (MU), that is, the moduli of their scalar products are equal to zero, one, or $1∕\sqrt{d}$. Each of these sets will be called a MU constellation, and if four MU bases were to exist for $d=6$, they would give rise to 35 different MU constellations. Using a numerical minimization procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension 6.

• Received 14 August 2008

DOI:https://doi.org/10.1103/PhysRevA.78.042312