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 This may sound outdated, but we show there is more to this connection than common sense may lead us to believe.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.

这项工作旨在明确两级量子系统的准备与其在希尔伯特空间中的相应描述（作为状态）之间的操作联系。这听起来可能已经过时了，但我们表明这种联系比常识可能让我们相信的更多。为了弥合这两个独立的领域——实际的实验室和状态空间——我们依赖于一个典型的数学对象：霍普夫纤维。我们通过简单的光学设置来说明这种连接在实践中是如何工作的。值得注意的是，这种光学设置还反映出使用两个图表来覆盖球体的必要性。换句话说，我们的实验实现反映了被视为平滑流形的球体的二维性。