Einstein Gyrogroup and Quantum Measurements in Spin Factor State Spaces

Einstein velocity addition endows the unit ball of Rn with the structure of a gyrocommutative gyrogroup together with the hyperbolic Hilbert-Klein Riemannian metric. First, we propose a self-contained derivation of these properties by rewriting Einstein addition in the Clifford algebra of the Euclidean space R^n , in which gyroautomorphisms are given by spinors. Then, our main result shows that the Einstein gyrogroup arises naturally in the context of quantum measurements in real spin factors. By embedding the state space of the Jordan algebra R ⊕ R^n into a multipartite rebit system realized as a tensor product of the algebra of 2 × 2 real symmetric matrices, we prove that Lüders state-update operations induce a gyrogroup morphism from the unit ball endowed with Einstein addition to a gyrogroup of density matrices. This extends known results in the low-dimensional case for the rebit and the qubit, and confirms that relativistic velocities and quantum state-updates in real spin factor share a common algebraic structure in arbitrary dimensions.