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 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.