If 1=2+3, then 1=2.3: Bell states, finite groups, and mutually unbiased bases, a unifying approach

Abstract

We study the relationship between Bell states, finite groups and complete sets of bases. We show how to obtain a set of N+1 bases in which Bell states are invariant. They generalize the X, Y and Z qubit bases and are associated to groups of unitary transformations that generalize the sigma operators of Pauli. When the dimension N is a prime power, we derive (in agreement with well-known results) a set of mutually unbiased bases. We show how they can be expressed in terms of the (operations of the) associated finite field of N elements.