Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions

Abstract

Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of (N + 1) mutually unbiased bases in Hilbert spaces of prime dimension N is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of on finite phase space implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime is also a finite field.