⚛️ quantum physics

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

This paper introduces a "solid spin Wigner function" based on an SU(3) representation to effectively model noisy spin-1/2 ensembles that escape the symmetric subspace, reducing the description to three interpretable parameters visualized on a solid ball rather than a hollow sphere.

Original authors: Andrew Kolmer Forbes

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Andrew Kolmer Forbes

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Noisy" Problem

Imagine you have a huge choir of N singers (these are your "spin-1/2 particles").

The Perfect Scenario: If everyone sings in perfect unison, they act like a single, giant super-voice. In physics, we call this the "symmetric subspace." We have a great map to visualize this: the SU(2) Wigner Function. Think of this map as a hollow beach ball. Every point on the surface represents a specific way the choir can sing together. It's beautiful, simple, and easy to draw.
The Real-World Problem: In the real world, things get messy. A singer might cough, lose their voice, or get distracted (this is "noise"). When this happens, the choir stops singing in perfect unison. They start singing in different, chaotic ways.
- The Issue: The old "hollow beach ball" map only works for the perfect choir. Once the noise starts, the singers drift off the surface of the ball and into the messy space inside. The old map becomes useless because it has no way to show what's happening in the "messy middle."

The Solution: Building a "Solid Ball" Map

The author, Andrew Kolmer Forbes, asks: "How do we make a new map that can show both the perfect singing (on the surface) and the messy noise (inside)?"

His answer is to stop thinking of the choir as a single group and start thinking of them as a 3D object.

1. The New Group: SU(3)

Instead of using the simple rules of the "hollow ball" (SU(2)), he uses a more complex set of rules called SU(3).

Analogy: Imagine the hollow ball is a 2D surface. SU(3) is like inflating that ball until it becomes a solid, 3D jelly ball.
Now, we have a surface (the perfect choir) and the entire volume inside the jelly (the noisy, imperfect choir).

2. The Magic Trick: The "Vectorized" Space

To make this math work, the author performs a clever trick. He takes the messy, noisy states and "promotes" them.

Analogy: Imagine you have a blurry, mixed-up photo of the choir. Usually, you can't tell who is who. The author takes that blurry photo and turns it into a crisp, high-definition 3D model where every singer is a distinct point in space.
This allows him to treat the messy noise as if it were a "pure" state in this new, larger 3D space.

3. The Result: The "Solid Spin Wigner Function"

The final result is a new way to visualize the quantum state.

The Old Way: A point on a hollow sphere.
The New Way: A point inside a solid ball.

The map now has three coordinates to describe the state:

Latitude (Polar Angle): Where are we on the surface? (North pole vs. South pole).
Longitude (Azimuthal Angle): Which direction are we facing?
Depth (Radial Component): This is the new part. How deep inside the ball is the point?
- The Surface (Depth = 1): Represents the "perfect" choir (no noise).
- The Center (Depth = 0): Represents the "maximally messy" choir (total noise).
- In Between: Represents a choir that is partially noisy.

Why is this useful?

1. Visualizing Noise:
In the old map, noise was invisible or impossible to draw. In this new "Solid Ball" map, you can literally watch a state "drift" from the surface toward the center as noise increases.

Example: If you have a "GHZ state" (a very fragile, quantum superposition), the map shows it as a bright spot on the surface. As noise hits, that spot shrinks and sinks toward the center of the ball, turning "washed out" and less distinct.

2. The "Radial" Component:
The author realized that the "messiness" of the noise could be described by a single number: how far from the center are we?

This turns a complex math problem with many variables into a simple 3D picture. You can look at a cross-section of the ball (like slicing an orange) to see exactly how the noise is affecting the system.

The Catch (The "Negativity" Issue)

There is one weird quirk. In quantum mechanics, these maps often have "negative" values (like negative probability), which is a sign of "quantumness."

In this new solid ball map, the author finds negative values appearing in the "depth" direction.
The Warning: He admits that some of this negativity might be an artifact of the math (because we invented a new group, SU(3), that doesn't naturally exist in the physical system). It's like drawing a map of a city using a grid system that doesn't quite match the streets; the map is useful, but you have to be careful not to take every line literally.

Summary

The Problem: Old maps for quantum spins only work for perfect, noise-free systems. Real systems get noisy and break the map.
The Fix: The author built a new map using a larger mathematical group (SU(3)).
The Visualization: Instead of a hollow sphere, the state is now a point inside a solid ball.
- Surface: Perfect order.
- Center: Total chaos/noise.
- Inside: The messy middle ground.
The Benefit: Scientists can now visualize and calculate how noise destroys quantum states by watching them "sink" from the surface to the center of the ball.

It's like upgrading from a flat map of the Earth's surface to a 3D globe that shows not just where you are, but how deep underground the "noise" has burrowed.

1. Problem Statement

The standard SU(2) Wigner function is a powerful tool for visualizing quantum states of spin- $J$ systems, representing them as real-valued functions on the surface of a 2-sphere (the Bloch sphere). This representation is valid when the system's dynamics are restricted to a single SU(2) irreducible representation (irrep), such as the symmetric subspace of $N$ spin-1/2 particles where $J = N/2$ .

However, physically relevant noise sources (e.g., spontaneous emission, incoherent optical pumping, depolarizing) are typically local. These local interactions break the permutation symmetry of the ensemble, transferring the quantum state out of the initial symmetric subspace ( $J=N/2$ ) into a larger Hilbert space containing multiple SU(2) irreps with varying total angular momenta $J$ .

The Limitation: The standard SU(2) Wigner function cannot describe states outside a single irrep.
The Naïve Solution: Constructing a Wigner function for the full $2^N$ dimensional Hilbert space (as a product of individual spin-1/2 Wigner functions) results in a $2N$ -dimensional parameter space, which is computationally intractable and lacks intuitive visualization.

The paper addresses the need for a phase-space representation that can handle noisy spin ensembles undergoing local noise while maintaining a low-dimensional, visualizable structure.

2. Methodology

The author proposes a novel approach by embedding the noisy spin ensemble into a larger group structure, specifically SU(3), rather than treating the system as a simple product of spins.

A. The Collective State Basis and Vectorization

Symmetric Noise: The work assumes local noise preserves permutation symmetry, restricting dynamics to the collective state basis. This basis spans the space of permutationally symmetric operators.
Vectorization: The author defines a mapping from the space of collective operators $\mathcal{C}$ (where mixed states exist due to degeneracy) to a vectorized space $\mathcal{V}$ . In $\mathcal{V}$ , the collective states are mapped to pure states $|J, M\rangle$ in a direct sum of SU(2) irreps:
$\mathcal{V} = \bigoplus_{J} H_J$
This mapping removes degeneracy labels and allows for the definition of jump operators that transfer population between different $J$ sectors (e.g., atom loss), which are difficult to define strictly within the operator algebra of $\mathcal{C}$ .

B. Isomorphism to SU(3)

The vectorized space $\mathcal{V}$ is shown to be isomorphic to the $(N, 0)$ irreducible representation of SU(3).
The mapping connects the quantum numbers of the collective state basis $(J, M, N)$ to the weight labels of the SU(3) irrep $(\nu_1, \nu_2, \nu_3)$ and the Dynkin indices $(\lambda, \mu) = (N, 0)$ .
Crucially, the SU(3) group contains three SU(2) subgroups. The author selects the {23} subgroup to represent the physical SU(2) rotations of the spin ensemble, while the {12} subgroup is used to encode the distribution of population across different $J$ sectors (the "radial" degree of freedom).

C. Construction of the Solid Spin Wigner Function

Using the Moyal quantization formalism and the Stratonovich-Weyl (SW) conditions, an SU(3) Wigner function is constructed on the coset manifold SU(3)/U(2).
Parameter Reduction: Physical constraints (specifically, the lack of coupling between different $J$ irreps in the collective basis) eliminate one of the four real parameters usually required for the SU(3)/U(2) manifold.
The Resulting Parameters: The Wigner function depends on only three real parameters:
1. Polar angle ( $\theta$ ): Standard spherical coordinate.
2. Azimuthal angle ( $\phi$ ): Standard spherical coordinate.
3. Radial component ( $r$ ): A new parameter derived from the angle $\beta$ associated with rotations in the {12} subgroup. It encodes the population distribution across different total angular momentum $J$ sectors.

3. Key Contributions

The Solid Spin Wigner Function: The primary contribution is the definition of a Wigner function visualized on a solid ball (unit radius) rather than a hollow sphere. The radius $r$ represents the "depth" of the state into the Hilbert space, corresponding to the mixture of different SU(2) irreps.
Dimensionality Reduction: The method reduces the description of a noisy $N$ -spin system from an exponentially large space to a 3-dimensional continuous manifold, making it tractable for simulation and visualization.
Kernel Derivation: The author derives an explicit analytical expression for the Wigner kernel $\hat{\omega}(\theta, \phi, r)$ . This kernel is expressed as a sum over SU(2) spherical tensors and spherical harmonics, weighted by a radial function $R_{J,k}(r)$ involving Legendre polynomials.
Handling Local Noise: The framework naturally incorporates local noise channels (like atom loss) that change the total angular momentum $J$ , which the standard SU(2) Wigner function cannot do.

4. Results and Visualizations

Visualization of States:
- Symmetric States ( $J=N/2$ ): These states are concentrated near the surface of the ball ( $r \approx 1$ ), recovering the behavior of the standard SU(2) Wigner function on the sphere.
- Lower $J$ States: States with lower total angular momentum are concentrated toward the center of the ball ( $r \approx 0$ ).
- Maximally Mixed States: The paper analyzes the maximally mixed state of $N$ spins. Unlike the SU(3) maximally mixed state (which is constant), the physical maximally mixed state of spins exhibits a non-uniform distribution in the solid ball, reflecting the degeneracy weights of the collective basis.
Noise Dynamics:
- Simulations of atom loss show the Wigner function "drifting" from the surface toward the center of the ball as $J$ decreases.
- Wigner Negativity: The paper observes that noise tends to wash out the negative regions of the Wigner function (a signature of non-classicality). However, it notes that the solid spin Wigner function exhibits negativity even in "classical" states (like coherent states) due to the artificial group actions (SU(3) rotations) introduced by the embedding. This negativity is interpreted as a mathematical artifact of satisfying the traciality condition rather than a direct measure of quantumness in the physical system.

5. Significance and Future Directions

New Visualization Paradigm: The "Solid Spin Wigner Function" provides an intuitive geometric picture for complex, noisy many-body quantum systems, bridging the gap between simple spin models and full many-body dynamics.
Computational Efficiency: By mapping the problem to a 3D manifold, it offers a potential pathway for more efficient numerical simulations of noisy spin ensembles compared to full density matrix evolution.
Theoretical Implications: The work highlights the trade-offs in defining Wigner functions for systems with broken symmetries. It suggests that while SU(3) is suitable for atom loss (half-integer jumps), other noise channels (integer jumps) might require different Lie algebras (e.g., SO(5)) for a more natural representation.
Quantum Resources: The paper contributes to the ongoing understanding of Wigner negativity as a resource, clarifying its behavior in finite-dimensional systems under local noise.

In summary, Forbes successfully generalizes the Wigner function formalism to noisy spin ensembles by leveraging the geometry of SU(3), creating a "solid ball" representation that captures both the angular orientation and the "purity" (angular momentum distribution) of the quantum state.