Spin-entangled Squeezed State on a Bloch Four-hyperboloid

The Bloch hyperboloid $H^2$ underlies the quantum geometry of the original $SO(2,1)$ squeezed states. In \cite{Hasebe-2019}, 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.