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

• URL: http://arxiv.org/abs/2508.16691v1
• Date: Thu, 21 Aug 2025 21:08:52 GMT
• Title: From Rotations to Unitaries: Reversible Quantum Processes and the Emergence of the $SU(2)-SO(3)$ Isomorphism
• Authors: V. G. Valle, B. F. Rizzuti,
• Abstract summary: We reconstruct the well-known (quasi-)isomorphism between the groups $SU(2)$ and $SO(3)$.<n>Our approach reveals how this group-theoretic structure naturally emerges from physical constraints.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present an operational reconstruction of the well-known (quasi-)isomorphism 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 CPTP 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.

Related papers

• Topological Preparation of Non-Stabilizer States and Clifford Evolution in $SU(2)_1$ Chern-Simons Theory [0.0]
  We develop a framework for preparing families of non-stabilizer states, and computing their entanglement entropies.<n>Using the Kac-Moody algebra, we construct Pauli and Clifford operators as path integrals over 3-manifolds with Wilson loop insertions.<n>Our results extend existing topological constructions for stabilizer states to include families of non-stabilizer states.
  arXiv  Detail & Related papers  (2025-10-16T18:29:50Z)
• Topological order in symmetric blockade structures [0.0]
  We study two- and three-dimensional structures of two-level systems that interact via a simple blockade potential.<n>Our construction is based on the implementation of a local symmetry on the microscopic level in a system with only two-body interactions.
  arXiv  Detail & Related papers  (2025-03-21T13:20:09Z)
• Doubly Quantum Mechanics [0.0]
  We develop the formalism for spin-$frac12$ measurements by promoting the group of spatial rotations $SU(2)$ to the quantum group $SU_q(2)$.<n>We find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of $SU_q(2)$.
  arXiv  Detail & Related papers  (2024-12-08T17:10:35Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Complex dynamics approach to dynamical quantum phase transitions: the Potts model [0.0]
  This paper introduces complex dynamics methods to study dynamical quantum phase transitions in the one- and two-dimensional quantum 3-state Potts model. We show that special boundary conditions can alter the nature of the transitions, and verify the claim for the one-dimensional system by transfer matrix calculations. Our approach can be extended to multi-variable problems, higher dimensions, and approximate RG transformations expressed as rational functions.
  arXiv  Detail & Related papers  (2023-08-28T18:26:41Z)
• Quantum entanglement and contextuality with complexifications of $E_8$ root system [55.2480439325792]
  The Witting configuration with 40 complex rays was suggested as a possible reformulation of Penrose model with two spin-3/2 systems based on geometry of dodecahedron.<n>An analysis of properties of suggested configuration of quantum states is provided using many analogies with properties of Witting configuration.
  arXiv  Detail & Related papers  (2022-10-27T11:23:12Z)
• Non-Gaussian superradiant transition via three-body ultrastrong coupling [62.997667081978825]
  We introduce a class of quantum optical Hamiltonian characterized by three-body couplings. We propose a circuit-QED scheme based on state-of-the-art technology that implements the considered model.
  arXiv  Detail & Related papers  (2022-04-07T15:39:21Z)
• Quantum reference frames: derivation of perspective-dependent descriptions via a perspective-neutral structure [0.0]
  We develop a symmetry-inspired approach to describe physics from the perspective of quantum reference frames. We show that the operationally meaningful perspective dependent descriptions are given by Darboux coordinates on the constraint surface. We conclude by constructing a quantum perspective neutral structure, via which we can derive and change perspective dependent descriptions.
  arXiv  Detail & Related papers  (2021-09-04T18:09:07Z)
• Realization of arbitrary doubly-controlled quantum phase gates [62.997667081978825]
  We introduce a high-fidelity gate set inspired by a proposal for near-term quantum advantage in optimization problems. By orchestrating coherent, multi-level control over three transmon qutrits, we synthesize a family of deterministic, continuous-angle quantum phase gates acting in the natural three-qubit computational basis.
  arXiv  Detail & Related papers  (2021-08-03T17:49:09Z)
• Unraveling the topology of dissipative quantum systems [58.720142291102135]
  We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hamiltonians.
  arXiv  Detail & Related papers  (2020-07-12T11:26:02Z)
• State preparation and measurement in a quantum simulation of the O(3) sigma model [65.01359242860215]
  We show that fixed points of the non-linear O(3) sigma model can be reproduced near a quantum phase transition of a spin model with just two qubits per lattice site. We apply Trotter methods to obtain results for the complexity of adiabatic ground state preparation in both the weak-coupling and quantum-critical regimes. We present and analyze a quantum algorithm based on non-unitary randomized simulation methods.
  arXiv  Detail & Related papers  (2020-06-28T23:44:12Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.