The algebraic structure of Mutually Unbiased Bases

Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in Copf
^d
are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that sets of complete MUBs only exist in Copf
^d
if a projective plane of size d also exists. We investigate the structure of MUBs using two algebraic tools: relation algebras and group rings. We construct two relation algebras from MUBs and compare these to relation algebras constructed from projective planes. We show several examples of complete sets of MUBs in Copf
^d
, that when considered as elements of a group ring form a commutative monoid. We conjecture that complete sets of MUBs will always form a monoid if the appropriate group ring is chosen.