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arXiv:2511.13266 · Journal of the Optical Society of America A

Antisymmetric Mueller generator as the universal origin of geometric phase in classical polarization and quantum two-level systems

José J. Gil

Published Jan 30, 2026• 1 citation
arXivHTMLPDF

Abstract

We show that the antisymmetric Mueller generator provides a universal algebraic kernel for geometric phase in classical polarization optics and in quantum two-level systems. For any ideal retarder, the antisymmetric 3×3 block of its Mueller matrix (the antisymmetric generator of the adjoint SU(2) action on the Stokes vector) encodes the angular-velocity vector that drives the tangential motion on the Poincaré sphere and fully determines the Pancharatnam–Berry phase, while the symmetric block is geometrically neutral. The same antisymmetric generator governs the evolution of pure qubit states on the Bloch sphere. This unified viewpoint yields operational criteria to identify and control geometric-phase contributions from measured Mueller matrices and from qubit process tomography.

DOI
10.1364/josaa.586965

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We show that the antisymmetric Mueller generator provides a universal algebraic kernel for geometric phase in classical polarization optics and in quantum two-level systems. For any ideal retarder, the antisymmetric 3×3 block of its Mueller matrix (the antisymmetric generator of the adjoint SU(2) action on the Stokes vector) encodes the angular-velocity vector that drives the tangential motion on the Poincaré sphere and fully determines the Pancharatnam–Berry phase, while the symmetric block is geometrically neutral. The same antisymmetric generator governs the evolution of pure qubit states on the Bloch sphere. This unified viewpoint yields operational criteria to identify and control geometric-phase contributions from measured Mueller matrices and from qubit process tomography.
We identify, for a general physically realizable Mueller transformation, the only intrinsic geometricphase structure that can be assigned to it in an invariant manner: the retarding part of the characteristic pure component selected by the characteristic decomposition, which defines a canonical holonomic content. A Mueller matrix does not, in general, determine a unique observed interferometric (Pancharatnam) geometric phase, since the latter depends on the specific physical realization of the transformation and on the interferometric readout. The remaining characteristic layers may modify the measured complex visibility, and even its observed argument through convex averaging, but they do not define a unique geometric holonomy of their own. We further establish the quantum analogue for open two-level dynamics within the Choi representation.
We identify, for a general physically realizable Mueller transformation, the only intrinsic geometric-phase structure that can be assigned to it in an invariant manner: the retarding part of the characteristic pure component selected by the characteristic decomposition, which defines a canonical holonomic content. A Mueller matrix does not, in general, determine a unique observed interferometric (Pancharatnam) geometric phase, since the latter depends on the specific physical realization of the transformation and on the interferometric readout. The remaining characteristic layers may modify the measured complex visibility, and even its observed argument through convex averaging, but they do not define a unique geometric holonomy of their own. We further establish the quantum analog for open two-level dynamics within the Choi representation.
⟨ ⟩ The qang project describes qubits by measurement—level quantities, the qg values qgP = P [1]. This short note states two classical results in those units, implemented and checked in the qang library (pip install qang, version 0.6.0). First, a polarization state of light is a qubit: the normalized Stokes parameters are exactly the qg values, the degree of polarization P is the length of the qg vector, and a lossless Mueller matrix is the qg gate rule qg′ = qgU†PU . Malus's law becomes I = I0(1 + qgZ cos 2θ + qgX sin 2θ)/2 for any input state. Second, the — geometric phase of a closed loop on the Bloch (Poincaré) sphere is minus half the enclosed solid angle, which is Stokes' theorem for the Berry curvature. For a loop at constant qgZ the phase is (qgZ 1)/2 of a turn: it is fixed by qgZ alone. Nothing here is new physics. The contribution is the translation into qang's units and a set of 26 numerical checks, with the phase computed three independent ways.
Geometric Phase in Quantum Mechanics is generally formulated entirely in terms of geometric structure of the Complex Hilbert Space. We will exploit this fact in case of mixed states for three level open systems undergoing depolarization using the eight di- mensional Poincare sphere and non unit vector rays in H3 within the limit of pure state approach may be found to be in agreement with the Pancharatnam Phase, Berry Phase and Aharonov-Anandan Phase. We will consider the Depolarization of the Three di- mensional fields by introducing the stokes parameters and by redefining the definition of Degree of Polarization using the SU(3) algebra to describe the quantum light depolariza- tion of Non-paraxial beams by minimal reservoir coupling models modeled by Lindblad Master equations and connect the solution of that with the geometric phase for systems undergoing depolarization.
By expanding the density matrix of the open system in terms of Gell-mann matrix in a three-level system, we parameterize coefficients of expansion by some azimuthal angles and find an identity mapping of the density matrices onto interior points of the unit Poincaré sphere. Thus, the relations between the points on the unit Poincaré sphere and wave functions are extended to connect the interior points in the sphere with the nonunit vector rays corresponding to an open system in complex Hilbert space. Thus,the geometric phases for the open system are proposed to be observed by the nonunit vector rays,where the geometric phase of the pure state is the limiting case of our definition. The results show that this geometric phase merely with duplicate three-dimensional Hilbert projection space geometry structure related, has nothing to do with the open system concrete evolution way; and it depends on population inversion and is a slippy and single-value curve of Bloch radius. Therefore, the mixed state of open system retains indeed a memory of its motion in the form of a geometric phase factor.