# Phase Space Insights: Wigner Functions for Qubits and Beyond
## 2 Wigner Function for a Spin \(S\)
The map is established as  
\[W_{A}(\theta,\phi)=\text{Tr}[\hat{A}\,\hat{w}(\theta,\phi)], \tag{4}\]  
where \(\hat{w}(\theta,\phi)\) is a Stratonovich-Weyl quantizer [19]:  
\[\hat{w}(\theta,\phi)=\sqrt{\frac{4\pi}{2S+1}}\sum_{K=0}^{2S}\sum_{q=-K}^{K} \hat{T}_{Kq}^{S}\,Y_{Kq}^{\ast}(\theta,\phi)=\hat{w}^{\dagger}(\theta,\phi). \tag{5}\]  
Here, \(Y_{Kq}(\theta,\phi)\) are the spherical harmonics and \(\hat{T}_{Kq}^{S}\) are the irreducible tensor operators [84; 85; 86]  
\[\hat{T}_{Kq}^{S}=\sqrt{\frac{2K+1}{2S+1}}\sum_{m,m^{\prime}=-S}^{S}C_{Sm,Kq}^{ sm^{\prime}}\ket{S,m^{\prime}}\bra{S,m}, \tag{6}\]  
with \(C_{S_{1}m_{1},S_{2}m_{2}}^{Sm}\) denoting the Clebsch-Gordan coefficients that couple a spin \(S_{1}\) and a spin \(S_{2}\) to a total spin \(S\). They are nonzero only when the standard angular momentum coupling rules hold: \(m_{1}+m_{2}=m\) and \(\ket{S_{1}-S_{2}}\leq S\leq S_{1}+S_{2}\). According to the properties of the Clebsch-Gordan coefficients, \(K\) takes the values \(0,1,2,\dots,2S\), and \(-K\leq q\leq K\), giving rise to \((2S+1)^{2}\) operators that constitute a basis for the space of linear operators acting on \(\mathcal{H}_{S}\). This is guaranteed by the property  
\[\text{Tr}\left(\hat{T}_{Kq}^{S}\hat{T}_{Kq^{\prime}}^{S*n}\right)=\delta_{SS^{ \prime}}\delta_{KK^{\prime}}\delta_{qq^{\prime}}. \tag{7}\]  
These operators are, in general, non-Hermitian. However, due to their symmetry properties, they satisfy the relation \(\hat{T}_{Kq}^{S\dagger}=(-1)^{q}\)\(\hat{T}_{K-q}^{S}\) for every fixed \(S\). Most importantly, they have the correct transformation properties under SU(2) transformations: for \(R(\Theta)\in\text{SU(2)}\), we have  
\[R(\Theta)\,\hat{T}_{Kq}^{S}\,R^{\dagger}(\Theta)=\sum_{q^{\prime}}D_{q^{ \prime}q}^{S}(\Theta)\,\hat{T}_{Kq^{\prime}}^{S}, \tag{8}\]  
where \(D_{q^{\prime}q}^{S}(\Theta)\) stands for the Wigner \(D\)-function [86] and we have used the compact notation \(\Theta=(\phi,\theta,\psi)\) for the three Euler angles characterizing the rotation. The kernel \(\hat{w}(\theta,\phi)\) is normalized as  
\[\text{Tr}[\hat{w}(\theta,\phi)]=1,\qquad\frac{2S+1}{4\pi}\int_{\mathcal{S}_{ 2}}d\Omega\,\hat{w}(\theta,\phi)=\mathds{1}, \tag{9}\]  
with \(d\Omega=\sin\theta d\theta d\phi\) as the invariant measure on the sphere \(\mathcal{S}_{2}\). All this ensures that the symbol of any (trace class) operator \(\hat{A}\) is covariant under rotations and provides the important overlap relation  
\[\frac{2S+1}{4\pi}\int_{\mathcal{S}_{2}}d\Omega\,W_{A}(\theta,\phi)W_{B}( \theta,\phi)=\text{Tr}(\hat{A}\hat{B}). \tag{10}\]  
As required, the map is invertible, so the operator \(\hat{A}\) can be reconstructed from its symbol via  
\[\hat{S}_{x}=\tfrac{1}{2}(\hat{a}_{1}\hat{a}_{2}^{\dagger}+\hat{a}_{1}^{\dagger }\hat{a}_{2}),\qquad\hat{S}_{y}=\tfrac{i}{2}(\hat{a}_{1}\hat{a}_{2}^{\dagger} -\hat{a}_{1}^{\dagger}\hat{a}_{2}),\qquad\hat{S}_{z}=\tfrac{1}{2}(\hat{a}_{1} ^{\dagger}\hat{a}_{1}-\hat{a}_{2}^{\dagger}\hat{a}_{2}), \tag{2}\]0  
Given the properties of the irreducible tensors, any operator can be expanded as  
\[\hat{S}_{x}=\tfrac{1}{2}(\hat{a}_{1}\hat{a}_{2}^{\dagger}+\hat{a}_{1}^{\dagger }\hat{a}_{2}),\qquad\hat{S}_{y}=\tfrac{i}{2}(\hat{a}_{1}\hat{a}_{2}^{\dagger} -\hat{a}_{1}^{\dagger}\hat{a}_{2}),\qquad\hat{S}_{z}=\tfrac{1}{2}(\hat{a}_{1} ^{\dagger}\hat{a}_{1}-\hat{a}_{2}^{\dagger}\hat{a}_{2}), \tag{2}\]1with \(A_{Kq}=\text{Tr}(\hat{A}_{Kq}\hat{T}_{Kq}^{\otimes\dagger})\).