Star product for qubit states in phase space and star exponentials

TL;DR

This paper develops a phase space framework for qubit systems using star products and exponentials, connecting operator algebra, quantum dynamics, and path integrals on the sphere.

Contribution

It introduces a deformation quantization on the sphere for qubits, linking star products, star exponentials, and coherent-state path integrals.

Findings

Star product reproduces quaternionic operator algebra.

Quantum dynamics expressed via star exponentials of Hamiltonians.

Established equivalence between path integral and algebraic star exponential formulations.

Abstract

In this paper, we formulate the phase space description of qubit systems using coadjoint orbits of $SU(2)$ and the Stratonovich-Weyl correspondence, yielding a deformation quantization on the sphere. The resulting star product reproduces the operator algebra of complexified quaternions and its antisymmetric part induces the Lie-Poisson structure associated with the Kirillov-Kostant-Souriau symplectic form. We show that quantum dynamics can be expressed entirely in phase space through star exponentials of Hamiltonian symbols, leading to an explicit representation of the propagator. Further, we establish the equivalence between the coherent-state path integral formulation on $S^2$ and the algebraic description in terms of star exponentials. Some examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included.