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

V. G. Valle, B. F. Rizzuti

TL;DR

The paper addresses reconstructing the $SU(2)$–$SO(3)$ two-to-one homomorphism from physical principles by linking rotations in physical space to unitary state transformations on qubits via the Bloch representation. Using an operational stance grounded in state preparation and reversible CPTP dynamics, it shows that a rotation $\\mathcal{R}(\\hat{n},\\alpha)$ corresponds to a unitary $\\mathcal{U}(\\hat{n},\\alpha)=e^{-i\\frac{\\alpha}{2}\\hat{n}\\cdot\\vec{\\sigma}}$ acting on states through $\\rho_{\\vec{r}'}=\\mathcal{U}\\rho_{\\vec{r}}\\mathcal{U}^*$, thereby realizing the mapping $\\phi: SO(3) \rightarrow SU(2)$. The authors then leverage a CPTP-inverse characterization to show that invertible physical maps reduce to unitary channels, which induces a rotation in physical space via the relation $\\mathcal{R}_{kj}=\\tfrac{1}{2}\mathrm{Tr}(\\mathcal{U}\\sigma_j\\mathcal{U}^*\\sigma_k)$, making the $SU(2)$–$SO(3)$ double cover explicit (with $\\Upsilon$ and $-\\Upsilon$ yielding the same rotation). The approach yields an operational, pedagogically accessible framework that ties preparation, reversible dynamics, and symmetry groups together, with prospects for extension to higher-dimensional systems and connections to recovery maps like the Petz map.

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.

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

TL;DR

The paper addresses reconstructing the – two-to-one homomorphism from physical principles by linking rotations in physical space to unitary state transformations on qubits via the Bloch representation. Using an operational stance grounded in state preparation and reversible CPTP dynamics, it shows that a rotation corresponds to a unitary acting on states through , thereby realizing the mapping . The authors then leverage a CPTP-inverse characterization to show that invertible physical maps reduce to unitary channels, which induces a rotation in physical space via the relation , making the – double cover explicit (with and yielding the same rotation). The approach yields an operational, pedagogically accessible framework that ties preparation, reversible dynamics, and symmetry groups together, with prospects for extension to higher-dimensional systems and connections to recovery maps like the Petz map.

Abstract

We present an operational reconstruction of the well-known two-to-one homomorphism between the groups and , 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 and tangible through experimentally meaningful procedures.

Paper Structure

This paper contains 7 sections, 3 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. From physical spaces to state spaces: $SO(3) \rightarrow SU(2)$
  3. Brief review on CPTP maps
  4. From state spaces to physical spaces: $SU(2) \rightarrow SO(3)$
  5. Conclusion and Perspectives
  6. Acknowledgments
  7. Appendix

Key Result

Theorem 2.1

Given a general rotation element $\mathcal{R}(\hat{n}, \alpha)\in SO(3)$. The state $\rho_{\vec{r}'}$, described by the vector $\vec{r}\,' = \mathcal{R}(\hat{n}, \alpha) \vec{r}$, can be calculated by applying a matrix $\mathcal{U}(\hat{n}, \alpha)$$\in SU(2)$ on the state $\rho_{\vec{r}}$, describe

Figures (2)

  • Figure 1: Diagram representing the allegedly relation between the groups $SO(3)$ and $SU(2)$.
  • Figure 2: Picture of $SU(2)$ and $SO(3)$ via the $su(2) \cong \mathds{D}$ bijection.

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 7.1