From Rotations to Unitaries: Reversible Quantum Processes and the Emergence of the SU(2)−SO(3) Homomorphism

Abstract We present an operational reconstruction of the well-known two-to-one homomorphism between the groups SU(2) and SO(3), grounded in the physical description of quantum state preparation and evolution. Starting from the connection between vectors in three-dimensional physical space and quantum states of two-level systems, we investigate how reversible transformations–modeled as completely positive and trace-preserving maps–give rise to a correspondence between spatial rotations and unitary operations. Our approach reveals how this group-theoretic structure naturally emerges from physical constraints, particularly the preservation of purity and reversibility in quantum processes. Beyond its theoretical relevance, the construction offers a pedagogically accessible framework for introducing core ideas in quantum mechanics and symmetry groups, making the abstract correspondence between SU(2) and SO(3) tangible through experimentally meaningful procedures.