Towards establishing a connection between two-level quantum systems and physical spaces

TL;DR

This paper explores the fundamental link between two-level quantum systems and their physical space representations using the Hopf fibration, supported by an optical experiment demonstrating the sphere's bi-dimensionality.

Contribution

It explicitly connects laboratory preparations of quantum states with their geometric description via the Hopf fibration, highlighting the manifold structure of the state space.

Findings

Optical setup illustrates the connection between quantum states and geometric space.

Demonstrates the necessity of two charts to cover the sphere in the quantum context.

Experimental realization reflects the bi-dimensionality of the sphere as a smooth manifold.

Abstract

This work seeks to make explicit the operational connection between the preparation of two-level quantum systems with their corresponding description (as states) in a Hilbert space. This may sound outdated, but we show there is more to this connection than common sense may lead us to believe. To bridge these two separated realms -- the actual laboratory and the space of states -- we rely on a paradigmatic mathematical object: the Hopf fibration. We illustrate how this connection works in practice with a simple optical setup. Remarkably, this optical setup also reflects the necessity of using two charts to cover a sphere. Put another way, our experimental realization reflects the bi-dimensionality of a sphere seen as a smooth manifold.