Qubit States Described in Phase Space

Jasel Berra-Montiel and colleagues present a method using coadjoint orbits and the Stratonovich-Weyl correspondence to represent qubit dynamics entirely within phase space. The method connects coherent-state path integrals with an algebraic description using star exponentials, reproducing the operator algebra of complexified quaternions and potentially offering a new set of tools for understanding quantum phenomena. It provides an explicit representation of the propagator and clarifies the underlying Poisson structure governing these systems. Deformation quantization extends to cosmological and black hole spacetimes Quantum mechanics on phase space provides a conceptually and technically strong framework where classical and quantum structures are treated equally. Within this approach, quantum observables are represented by functions on a classical phase space, utilising a noncommutative associative product known as the star product, which encodes the operator algebra. This representation, central to the deformation quantization program, replaces canonical commutation relations with a deformation of the commutative algebra C∞(M) of smooth functions on a symplectic manifold M, such that the antisymmetric component of the star product reproduces the Poisson bracket Recent developments have extended these methods to curved or dynamical spacetimes, encompassing cosmological backgrounds and black-hole geometries. Phase-space techniques offer a natural language for describing quantum dynamics in gravitational environments. Applying this to finite-dimensional quantum systems necessitates a different geometric setting, as continuous variable quantum systems on flat phase spaces, M = R2n, are well understood, with the Moyal product providing an explicit realization of the star product. For spin systems, the algebra of observables is finite-dimensional and non-abelian, lacking global canonical coordinates. The coadjoint orbits of the underlying symmetry group provide a natural phase space. Considering a qubit, the relevant group is SU, and its nontrivial coadjoint orbits are two-dimensional spheres S2, carrying a canonical symplectic structure defined by the Kirillov, Kostant, Souriau (KKS) form. This construction equips S2 with a Poisson bracket inherited from the Lie algebra structure on su∗, providing a natural classical phase space for spin systems. Quantization on these curved phase spaces can be implemented through the Stratonovich, Weyl (SW) correspondence, establishing an equivariant isomorphism between operators acting on a finite-dimensional Hilbert space and functions on the coadjoint orbit. This correspondence allows one to define phase-space symbols and induces an associative star product on C∞(S2). Unlike formal deformation approaches defining the star product as an asymptotic formal power series in ħ without guaranteed convergence, the resulting star product on the sphere is exact and finite, reflecting the underlying matrix algebra structure. Its antisymmetric part defines a Moyal bracket which reproduces the Sun Life algebra and, upon suitable identification, induces the Poisson structure associated with the KKS symplectic form. This work explicitly constructs and analyzes the star product associated with the SW correspondence for qubit systems and investigates its role in the phase-space formulation of quantum dynamics. The star product provides a faithful realization of the operator algebra on M2(C) in terms of functions on the sphere, capturing the full noncommutative structure of the theory. Furthermore, it can be naturally interpreted as a phase-space realization of the algebra of complexified quaternions. The notion of star exponentials for Hamiltonian symbols on S2 is developed, yielding a natural phase-space representation of the unitary time-evolution operator. This allows the propagator to be expressed entirely in terms of phase-space quantities, leading to an exact formulation of quantum dynamics within the deformation quantization framework. These two descriptions, the algebraic formulation in terms of star exponentials and the geometric formulation in terms of path integrals, are equivalent and constitute two complementary realizations of quantum dynamics on curved phase space. This correspondence extends the well-known equivalence between Moyal quantization and Feynman path integrals in R2n to the setting of coadjoint orbits. The paper organizes itself as follows. Section 2 reviews the construction of the coadjoint orbits of the Lie group SU together with their symplectic structure. Section 3 constructs a star product on this symplectic manifold via the Stratonovich, Weyl correspondence. Section 4 analyzes the relation between star-exponential functions and quantum mechanical propagators, discussing their natural connection within the framework of Feynman’s path integral. Explicit examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included. Finally, Section 5 provides a summary and outlook. Finite-dimensional quantum systems usually do not admit a globally defined phase space of this form, unlike continuous-variable quantum systems where phase space is identified with the symplectic manifold R2n, endowed with its standard Poisson structure. The finite dimensionality of the observable algebra for spin systems prevents the use of global canonical position and momentum coordinates. However, a natural notion of phase space is obtained for finite dimensional quantum systems via the coadjoint orbit method, analysing the orbits of the coadjoint action of a Lie group on the dual of its Lie algebra. In the case of qubits, this construction identifies the phase space with a two-dimensional coadjoint orbit of SU, equipped with the Kirillov, Kostant, Souriau symplectic form, providing a well-defined Poisson structure and a natural setting for deformation quantization. Let su be the Lie algebra of the group SU, generated by Ji = 1/2σi, i = 1, 2, 3, where σi denotes the Pauli matrices. With this normalization, the generators satisfy the commutation relations [Ji, Jj] = iεijkJk, and provide the fundamental spin-1/2 representation of the Sun Life algebra. As a real algebra, su is three-dimensional and simple, meaning any element X ∈su can be written uniquely as X = xiJi, where xi ∈R and the Einstein summation convention over repeated indices is understood. The Killing form on su, given by κ(X, Y ) = −2tr (X, Y ), is non-degenerate and therefore induces a canonical identification between the Lie algebra and its dual, su ≃su∗, via the correspondence X 7→κ(X, ·). The adjoint action of SU on su is the smooth group homomorphism Ad: SU →Aut(su), defined by Adg(X) := d/dtg exp(tX)g−1 at t=0, where g ∈SU and X ∈su. This takes the explicit form Adg(X) = gXg−1. The adjoint action is a Lie algebra homomorphism from su into itself, preserving the Lie bracket Adg[X, Y ] = [AdgX, AdgY ]. The infinitesimal version of the adjoint action is the adjoint representation ad: su →End(su), given by adX(Y ) := [X, Y ], X, Y ∈su. This map is obtained by differentiating the adjoint action at the identity element of the group. More precisely, for any X, Y ∈su, adX(Y ) = d/dtAdexp(tX)(Y ) at t=0. The adjoint representation defines a homomorphism of Lie algebras, since [adX, adY ] = ad[X,Y ]. Let su∗ denote the dual Lie algebra of su. The coadjoint action Ad∗: SU →Aut(su∗), is defined as the dual of the adjoint action ⟨Ad∗ gξ, X⟩:= ⟨ξ, Adg−1X⟩, where X ∈su and ξ ∈su∗. Its infinitesimal version reads (ad∗ Xξ)(Y ) = −ξ([X, Y ]), X, Y ∈su. The coadjoint orbit of ξ ∈su is given by Oξ:= Ad∗ gξ: g ∈SU ⊂su∗. Coadjoint orbits are immersed homogeneous submanifolds of dual Lie algebras carrying canonical symplectic structures which play a fundamental role in Lie theory, symplectic geometry, and geometric quantization. To make the construction of the orbits explicit, let ξ ∈su∗. Since the Killing form is Ad-invariant, namely κ(AdgX, AdgY ) = κ(X, Y ), the identification of su with su∗ is also SU-equivariant. The linear map Φ: su →su∗ defined by Φ(X) = κ(X, ·) is not only an isomorphism of vector spaces, but also intertwines the adjoint and coadjoint actions, in the sense that Ad∗ gΦ(X) = Φ(AdgX). Consequently, under this identification the coadjoint and adjoint orbits are mapped into each other orbit by orbit. Therefore, the non-trivial coadjoint orbits of SU are two-dimensional spheres. Finally, a symplectic structure can be constructed by considering the coadjoint orbits of a Lie group acting on the dual space of its Lie algebra. For the adjoint orbit through a point X ∈su, the tangent space is given by TXOX:= {adY X: Y ∈su}. Using the SU-equivariant identification between su and su∗, an analogous expression holds for the coadjoint orbit through ξ ∈su∗, namely TξOξ:= {ad∗ Y ξ: Y ∈su}. Every coadjoint orbit carries a canonical symplectic structure, known as Kirillov, Kostant, Souriau (KKS) symplectic form, defined at ξ ∈Oξ by ωξ(ad∗ Xξ, ad∗ Y ξ) = ⟨ξ, [X, Y ]⟩, X, Y ∈su. The form ωξ is bilinear, antisymmetric, closed, and non-degenerate on the orbit, providing the coadjoint orbit Oξ with the structure of a symplectic manifold. Precise algebraic formulation of qubit dynamics via complexified quaternion reproduction The star product is exact and finite, representing a major advance over previous formal deformation approaches reliant on asymptotic series. Earlier methods yielded approximations, whereas this delivers a complete, convergent solution for describing qubit behaviour. This precise algebraic structure faithfully reproduces the operator algebra of complexified quaternions, offering a novel framework for understanding quantum systems. By formulating quantum dynamics entirely within phase space using coadjoint orbits of SU(2) and the Stratonovich-Weyl correspondence, a direct equivalence between path integral formulations and algebraic descriptions using star exponentials has been established. Geometric orbits provide a novel visualisation of simple qubit states Researchers are increasingly focused on translating the abstract mathematics of quantum mechanics into geometrical language, seeking a more intuitive understanding of how qubits, the building blocks of quantum computers, behave. This latest work, detailing a phase-space description using coadjoint orbits, offers a compelling alternative to traditional approaches centred on wave functions and operators. Although the current formulation is restricted to basic qubits, lacking demonstrated applicability to more complex quantum systems, its potential remains significant. Representing quantum behaviour using shapes, rather than traditional wave functions, offers a fundamentally different perspective for physicists. This alternative framing could unlock new ways to visualise and ultimately manipulate quantum information, even if scaling up to complex systems presents a future challenge.

Scientists have detailed a new geometrical framework representing qubits using shapes instead of conventional wave functions. This approach utilises coadjoint orbits and a technique called the Stratonovich-Weyl correspondence to describe quantum behaviour in phase space, offering a different visualisation method. This work establishes a complete phase-space description of qubit systems, utilising coadjoint orbits, geometrical shapes representing quantum states, and the Stratonovich-Weyl correspondence. By representing quantum dynamics entirely within this phase space, scientists have demonstrated an equivalence between calculating quantum behaviour using path integrals and algebraic methods involving ‘star exponentials’. The resulting algebraic structure accurately mirrors the behaviour of complexified quaternions, a mathematical extension of quaternions, and reveals an underlying Poisson structure governing these systems. The research successfully formulated a phase-space description of qubit systems using geometrical shapes called coadjoint orbits and the Stratonovich-Weyl correspondence. This provides an alternative to conventional methods of representing quantum behaviour, potentially offering new ways to visualise quantum information. Researchers demonstrated that quantum dynamics can be fully expressed within this phase space, establishing a link between path integral formulations and algebraic calculations using star exponentials. The resulting framework accurately reflects the mathematical properties of complexified quaternions and reveals an associated Poisson structure. 👉 More information 🗞 Star product for qubit states in phase space and star exponentials 🧠 ArXiv: https://arxiv.org/abs/2604.05170