Describing Symmetric Noise on SU(2) Systems using an SU(3) Wigner Function

In collective spin systems, states in the symmetric subspace can be represented as Wigner functions on a sphere, which are constructed from the group symmetry of the underlying irreducible representation (irrep) of SU(2). For ensembles of spins undergoing local noise, this representation no longer applies. To remedy this, we present a mapping from SU(2) systems undergoing local symmetric noise to an irrep of SU(3). This mapping makes use of the collective state subspace of N spin-1/2 particles, which is the space of all permutationally symmetric states (mixed and pure). By finding an isomorphism between this space and SU(3) irreps of type $(\lambda,0)$ we can describe the former using an SU(3) Wigner function. Further, by making use of the physical constraints on the collective state subspace, we are able to simplify the SU(3) Wigner function such that it only takes 3 real parameters as inputs. These parameters can then be interpreted as a polar, azimuthal, and radial component, leading to visualizations on a Bloch "ball" rather than a sphere. This mapping opens the door to investigating properties such as Wigner negativity for large spin systems undergoing local noise.