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

TL;DR

This paper presents a group-theoretical method to construct the maximum number of mutually unbiased bases in finite-dimensional Hilbert spaces of prime dimension, leveraging properties of the finite Heisenberg group and finite fields.

Contribution

It provides an alternative proof for the existence of N+1 mutually unbiased bases in prime-dimensional Hilbert spaces using group theory and finite field properties.

Findings

Maximal sets of mutually unbiased bases exist in prime dimensions.

The construction uses the finite Heisenberg group and SL(2,Z_N) actions.

The proof relies on the finite field structure of Z_N for prime N.

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 SL(2,Z_N) on finite phase space Z_N x Z_N implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime N, Z_N is also a finite field.