Mutually Unbiased Bases for Continuous Variables

TL;DR

This paper explores mutually unbiased bases for continuous variables, linking symplectic geometry and number theory, and identifies specific bases for one and two pairs of variables, revealing the golden ratio's role.

Contribution

It extends the concept of mutually unbiased bases to continuous variables and uncovers their geometric and number-theoretic properties, including explicit bases for multiple pairs.

Findings

Identified three mutually unbiased bases for a single pair of continuous variables.

Constructed five mutually unbiased bases for two pairs of continuous variables.

Discovered the golden ratio's appearance in the structure of these bases for two pairs.

Abstract

The concept of mutually unbiased bases is studied for N pairs of continuous variables. To find mutually unbiased bases reduces, for specific states related to the Heisenberg-Weyl group, to a problem of symplectic geometry. Given a single pair of continuous variables, three mutually unbiased bases are identified while five such bases are exhibited for two pairs of continuous variables. For N = 2, the golden ratio occurs in the definition of these mutually unbiased bases suggesting the relevance of number theory not only in the finite-dimensional setting.