Symplectic geometry of unbiasedness and critical points of a potential

TL;DR

This paper explores the symplectic geometric interpretation of algebraically unbiased projectors, linking their classification to critical points of a Laurent polynomial potential, and connecting it with mirror symmetry and Fano varieties.

Contribution

It introduces a symplectic geometric framework for classifying algebraically unbiased projectors and relates the moduli space to critical points of a Laurent polynomial potential.

Findings

The moduli space corresponds to critical points of a Laurent polynomial.

The Newton polytope of the polynomial is the Birkhoff polytope.

Mirror symmetry relates the polynomial to a Landau-Ginzburg potential for a Fano variety.

Abstract

The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential functions, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of double stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geometry of the variety with systems of unbiased projectors.