Cliffordtori and unbiased vectors

TL;DR

This paper investigates the intersection properties of Clifford tori in complex projective 2-space, providing new insights into the geometric aspects related to mutually unbiased bases in quantum information theory.

Contribution

It offers a detailed analysis of Clifford tori intersections in the simplest non-trivial case, with parametrization and exploration of higher dimensions, advancing understanding of unbiased vectors.

Findings

Clifford tori intersections are characterized in complex projective 2-space.

Parametrization of Clifford tori via unistochastic subset of Birkhoff's polytope.

Insights into higher-dimensional cases and their geometric properties.

Abstract

The existence problem for mutually unbiased bases is an unsolved problem in quantum information theory. A related question is whether every pair of bases admits vectors that are unbiased to both. Mathematically this translates to the question whether two Lagrangian Clifford tori intersect, and a body of results exists concerning it. These (deep!) results are however rather weak when viewed from the point of view of the first problem. We make a detailed study of how the intersections behave in the simplest non-trivial case, that of complex projective 2-space (the qutrit), for which the set of Clifford tori can be usefully parametrized by the unistochastic subset of Birkhoff's polytope. An interesting picture emerges. A foray into higher dimensions is included.