Mutually unbiased bases are complex projective 2-designs

Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d + 1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. In this paper we demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0, 1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}.