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arXiv:2603.15387

Using an SU(3)/U(2) Wigner Function to Represent Noisy Spin Ensembles

Andrew Kolmer Forbes

quant-phSubmitted Mar 16, 2026• Announced Mar 17, 2026• 0 citations
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Abstract

The SU(2) Wigner function represents a quantum state of a spin-$J$ as a real-valued function on the surface of a 2-sphere. For an ensemble of $N$ spin-1/2 particles, this representation is useful when the dynamics is restricted to a single SU(2) irrep, e.g., the symmetric subspace with $J=N/2$. Physically relevant noise sources tend to be local, such as spontaneous emission, depolarizing, and incoherent optical pumping, all of which transfer the state outside of the initial irrep, and as such the SU(2) Wigner function is no longer a useful representation. In this work, we address this issue by encoding a noisy spin ensemble in an SU(3) irrep, and evaluating the SU(3) Wigner function for that irrep. We find that physical constraints enforced by the noise eliminate all but three real parameters from the input to the Wigner function, which can then be interpreted as a polar, azimuthal, and radial component. This interpretation leads us to refer to the resulting Wigner function as the solid spin Wigner function, visualized on a solid ball rather than a hollow sphere.

Comments
19 pages, 6 figures

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The SU(2) Wigner function represents a quantum state of a spin-JJJ as a real-valued function on the surface of a 2-sphere. For an ensemble of NNN spin-1/2 particles, this representation is useful when the dynamics is restricted to a single SU(2) irrep, e.g., the symmetric subspace with J=N/2J=N/2J=N/2. Physically relevant noise sources tend to be local, such as spontaneous emission, depolarizing, and incoherent optical pumping, all of which transfer the state outside of the initial irrep, and as such the SU(2) Wigner function is no longer a useful representation. In this work, we address this issue by encoding a noisy spin ensemble in an SU(3) irrep, and evaluating the SU(3) Wigner function for that irrep. We find that physical constraints enforced by the noise eliminate all but three real parameters from the input to the Wigner function, which can then be interpreted as a polar, azimuthal, and radial component. This interpretation leads us to refer to the resulting Wigner function as the solid spin Wigner function, visualized on a solid ball rather than a hollow sphere.
Nonclassicality in quantum sensors can improve sensitivity, but often increases susceptibility to noise. Thus, modeling physically relevant noise sources and analyzing their effect on quantum metrology are both of importance to the field of quantum sensing. In this dissertation, I demonstrate that local noise sources, which are present in almost all many-spin systems, can be tractably modeled when assuming permutation symmetry of the noise, and we show that many common local noise sources can be mapped to a Fokker-Planck equation on quantum phase space. We apply this description of noise to study quantum sensing using noisy probe states and establish a previously unknown connection between loss of Fisher information for noisy states and discontinuities in the Fisher information as a function of a continuous family of measurements. Finally, we extend the phase space description of local noise by developing a new spin Wigner function, which we call the solid spin Wigner function. We develop this new Wigner function by noting that the permutationally symmetric operator space used to describe symmetric local noise resembles the symmetric subspace of an ensemble of SU(3) particles. Using this, it is possible to embed the SU(2) group structure of the spin ensemble within this larger group, and describe the system using an SU(3) Wigner function.
Bases for SU(3) irreps are constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions. The Weyl group is represented as the symmetric group of permutations of the particle coordinates of these spaces. Wigner functions for SU(3) are expressed as products of SU(2) Wigner functions and matrix elements of Weyl transformations. The constructions make explicit use of dual reductive pairs which are shown to be particularly relevant to problems in optics and quantum interferometry.
In 1981, Agarwal proposed a Wigner quasiprobability distribution function on the group SU(2) that serves to analyze two-particle spin states on a sphere. Recent work by our group has included the definition of an apparently distinct Wigner function on generic Lie groups whose natural range has the dimension of the group and serves for all square-integrable representations; for the SO(3) case this entails a three-dimensional ``meta-phase'' space. Both have the fundamental properties covariance and completeness. Here we show how the former is obtained as a restriction of the latter.
Phase space methods, particularly Wigner functions, provide intuitive tools for representing and analyzing quantum states. We focus on systems with SU(2) dynamical symmetry, which naturally describes spin and a wide range of two-mode quantum models. We present a unified phase space framework tailored to these systems, highlighting its broad applicability in quantum optics, metrology, and information. After reviewing the core SU(2) phase-space formalism, we apply it to states designed for optimal quantum sensing, where their nonclassical features are clearly revealed in the Wigner representation. We then extend the approach to systems with an indefinite number of excitations, introducing a generalized framework that captures correlations across multiple SU(2)-invariant subspaces. These results offer practical tools for understanding both theoretical and experimental developments in quantum science.
Phase-space representations as given by Wigner functions are a powerful tool for representing the quantum state and characterizing its time evolution in the case of infinite-dimensional quantum systems and have been widely used in quantum optics and beyond. However, much less is known for finite-dimensional quantum systems such as spin systems and we present a complete theory of Wigner functions for this finite-dimensional case. In particular, we provide a self-contained Wigner formalism for describing and predicting the time evolution of coupled spins which lends itself to visualizing the high-dimensional structure of multi-partite quantum states. We completely treat the case of an arbitrary number of spins 1/2, thereby establishing the equation of motion using Wigner functions. The explicit form of the time evolution is then calculated for up to three spins 1/2. The underlying physical principles of our Wigner representations are illustrated with multiple examples for coupled spin systems which are easily translatable to other experimental scenarios.
We present a surprisingly simple three-dimensional Bloch sphere representation of a qutrit, i.e., a single three-level quantum system. We start with a symmetric state of a two-qubit system and relate it to the spin-1 representation. Using this representation we associate each qutrit state with a three-dimensional vector a\mathbf{a}a and a metric tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ which satisfy $\mathbf{a}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{a}\ensuremath{\le}1$. This resembles the well known condition for qubit Bloch vectors in which case $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}=\mathbb{1}$. In our case the vector a\mathbf{a}a corresponds to spin-1 polarization, whereas the tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of polarization uncertainties. Alternatively, a\mathbf{a}a is a local Bloch vector of a symmetric two-qubit state and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of the corresponding correlation tensor.