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

TL;DR

This paper provides an operational derivation of the $SU(2)$ to $SO(3)$ homomorphism using physical quantum processes, illustrating how symmetry groups naturally emerge from reversible quantum state transformations.

Contribution

It offers a new, physically grounded reconstruction of the $SU(2)$ to $SO(3)$ correspondence based on quantum state preparation and evolution.

Findings

Reversible quantum processes induce the $SU(2)$ to $SO(3)$ homomorphism.

The framework connects physical quantum operations with abstract symmetry groups.

Pedagogical approach makes the $SU(2)$-$SO(3)$ relation accessible through experiments.

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.