Holonomy in the quantum navigation problem

B Hubicska,Z Muzsnay

doi:10.1007/s11128-019-2438-8

Abstract

In this paper, we investigate the holonomy structure of the Quantum Zermelo navigational problem introduced by Russel and Stepney (Phys Rev A 90:012303, 2014). We show that the curvature algebra describing infinitesimally the non-commutativity of parallel translation is infinite dimensional. Consequently, we obtain that the holonomy group cannot be a finite-dimensional group. These results demonstrate the deep difference between the Riemannian and Finslerian model and show that new phenomena occur.

Highlights

In recent years, more and more focus tends to the applications of Finsler geometry in several fields of natural sciences
From general relativity through wildfire spread and seismic ray modeling to quantum mechanics, a lot of different areas take advantage of the tools of Finsler geometry [15,18]
The appearance of Randers manifolds, which are special types of Finsler manifolds, in Quantum Information Processing (QIP) together with the method for realizing quantum computation through holonomy transformations proposed by Zanardi and Rasetti [23] served as a base motivation for us to start investigating Finslerian holonomy groups in a quantum mechanical setting

Summary

Introduction

More and more focus tends to the applications of Finsler geometry in several fields of natural sciences. This phenomenon occurs in different areas of physics such as string theory [11] and what before the middle of the twentieth century was not considered as a geometric theory: quantum mechanics. The appearance of Randers manifolds, which are special types of Finsler manifolds, in QIP together with the method for realizing quantum computation through holonomy transformations proposed by Zanardi and Rasetti [23] served as a base motivation for us to start investigating Finslerian holonomy groups in a quantum mechanical setting. We obtain that the holonomy group cannot be a finite-dimensional group This result demonstrates the radical difference between the Riemannian and Finslerian model and shows that with respect to the classical Riemannian settings, a new phenomenon occurs: the presence of an ineliminable force extends hugely the set of holonomy transformations and the set of states realizable from one given state

Randers model of quantum information processing

Finsler metric

Covariant derivative and parallel translation

Holonomy of quantum information processing in Randers model

Curvature algebra and holonomy

Conclusion