VOL. 83 | 2019 Symplectic geometry of unbiasedness and critical points of a potential
Alexey Bondal, Ilya Zhdanovskiy

Editor(s) Kentaro Hori, Changzheng Li, Si Li, Kyoji Saito

Adv. Stud. Pure Math., 2019: 1-18 (2019) DOI: 10.2969/aspm/08310001

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 function, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of doubly 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.

Information

Published: 1 January 2019
First available in Project Euclid: 26 December 2019

zbMATH: 07276137

Digital Object Identifier: 10.2969/aspm/08310001

Subjects:
Primary: 14J33 , 14J45 , 14M25 , 35Q56 , 53D12 , 53D20 , 53D37 , 81P45

Keywords: Fano variety , Lagrangian submanifold , Landau-Ginzburg potential , mirror symmetry , momentum map , Mutually unbiased bases , Newton polyhedra , quantum information , symplectic reduction , toric varieties

Rights: Copyright © 2019 Mathematical Society of Japan

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