The Bloch hyperboloid H ² underlies the quantum geometry of the original SO(2, 1) squeezed states. In Hasebe (2020 J. Phys. A: Math. Theor. 53 055303), the author utilized a non-compact 2nd Hopf map and a Bloch four-hyperboloid H ^2,2 to explore an SO(2, 3) extension of the squeezed states. In the present paper, we further pursue the idea to derive an SO(4, 1) version of squeezed vacuum based on the other Bloch four-hyperboloid H ⁴. We show that the obtained SO(4, 1) squeezed vacuum is a particular four-mode squeezed state not quite similar to the previous SO(2, 3) squeezed vacuum. In view of the Schwinger’s formulation of angular momentum, the SO(4, 1) squeezed vacuum is interpreted as a superposition of an infinite number of maximally entangled spin-pairs of all integer spins. We clarify basic properties of the SO(4, 1) squeezed vacuum, such as von Neumann entropy of spin entanglement, spin correlations and uncertainty relations with emphasis on their distinctions to the original SO(2, 1) case.

Bloch 双曲面H ²是原始SO (2, 1) 压缩态的量子几何结构的基础。在 Hasebe (2020 J. Phys. A: Math. Theor. 53 055303) 中，作者利用非紧致 2nd Hopf 图和 Bloch 四双曲面H ^2,2来探索SO (2, 3) 扩展挤压状态。在本文中，我们进一步追求基于另一个 Bloch 四双曲面H ^4推导出SO (4, 1) 版本的压缩真空的想法。我们表明，获得的SO (4, 1) 压缩真空是一种特殊的四模压缩状态，与之前的SO不太相似 (2, 3) 挤压真空。鉴于 Schwinger 的角动量公式，SO (4, 1) 压缩真空被解释为所有整数自旋的无限数量的最大纠缠自旋对的叠加。我们阐明了SO (4, 1) 压缩真空的基本性质，例如自旋纠缠的冯诺依曼熵、自旋相关性和不确定性关系，重点是它们与原始SO (2, 1) 情况的区别。