# Phase Space Insights: Wigner Functions for Qubits and Beyond
## 4 Generalized Wigner Function  
In two-mode systems, fluctuations in the number of excitations are generally inevitable. As described in (3), the corresponding phase space becomes then foliated into a series of nested spheres, with radii proportional to the excitation number--commonly referred to as Fock layers in photonic systems [104]. To address this structure, we must extend earlier approaches to encompass systems whose state spaces form a direct sum of distinct SU(2) irreps. Our strategy follows the framework introduced in ref. [63], which employs a natural generalization of irreducible tensor operators [85]; namely,  
\[\hat{T}^{SS^{\prime}}_{kq}=\sum_{m=-S}^{S}\sum_{m^{\prime}=-S^{\prime}}^{S^{ \prime}}\sqrt{\frac{2K+1}{2S^{\prime}+1}}C^{S^{\prime}m^{\prime}}_{Sm,Kq}\ket{ S^{\prime},m^{\prime}}\bra{S,m}. \tag{33}\]  
The range of the indices is still \(\ket{S-S^{\prime}}\leq K\leq\ket{S+S^{\prime}}\) and \(-K\leq q\leq K\). The tensors (33) also form an orthonormal basis  
\[\text{Tr}\Big{(}\hat{T}^{S_{1}S_{1}^{\prime}}_{k_{1}q_{1}}\hat{T}^{S_{2}S_{2} ^{\prime}\pmq}_{k_{2}q_{2}}\Big{)}=\delta_{S_{1}S_{2}^{\prime}}\delta_{S_{1}S_ {2}}\delta_{K_{1}K_{2}}\delta_{q_{1}q_{2}}, \tag{34}\]  
and transform under rotations as in (8). Figure 4: Density plots on the unit Bloch sphere of the Wigner functions for NOON states, with the indicated values of \(S\). The scale of the colormap is indicated at the right. In consequence, we can expand any operator as  
\[\hat{A}=\sum_{S,S^{\prime}=0,\frac{1}{2},1,\ldots}\sum_{K=|S-S^{\prime}|}^{S+S^{ \prime}}\sum_{\vec{q}=-K}^{K}A_{\vec{k}q}^{SS^{\prime}}\hat{T}_{\vec{k}q}^{SS^{ \prime}}. \tag{35}\]  
By redefining the summation indices as \(J=S+S^{\prime}\), \(q^{\prime}=S^{\prime}-S\) and applying the resummation identity  
\[\sum_{q^{\prime}=-J}^{J}\sum_{K=|q^{\prime}|}^{J}a_{q^{\prime}K}=\sum_{K=\{0, \frac{1}{2}\}}\sum_{q^{\prime}=-K}^{K}a_{q^{\prime}K}, \tag{36}\]  
where \(\{0,\frac{1}{2}\}\) indicates that \(K\) takes values starting from \(0\) for integer \(J\) and from \(\frac{1}{2}\) for half-integer \(J\), we obtain the following result:  
\[\hat{A}=\sum_{J=0,\frac{1}{2},1,\ldots}\sum_{K=\{0,\frac{1}{2}\}}\sum_{q,q^{ \prime}=-K}^{K}A_{\vec{k}q}^{\frac{J+q^{\prime}}{2}\frac{1-q^{\prime}}{2}} \hat{T}_{\vec{k}q}^{\frac{1+q^{\prime}}{2}\frac{1-q^{\prime}}{2}}\equiv\sum_ {J=0,\frac{1}{2},1,\ldots}\hat{A}_{J}. \tag{37}\]  
Now, we can define the corresponding Stratanovitch-Weyl quantizer as  
\[\hat{\omega}_{J}(\Theta)=\sum_{K=\{0,\frac{1}{2}\}}\sum_{q,q^{\prime}=-K}^{K} \sqrt{\frac{2K+1}{J+1}}D_{qq^{\prime}}^{K}(\Theta)\hat{T}_{\vec{k}q}^{\frac{1 +q^{\prime}}{2}\frac{1-q^{\prime}}{2}}. \tag{38}\]  
This quantizer possesses all the desired properties [105] and can be used as an invertible mapping of arbitrary operator into \(c\)-number functions. The symbol of the operator \(\hat{A}\) can be consequently defined as  
\[W_{A}^{J}(\Theta)=\operatorname{Tr}[\hat{A}\,\hat{\omega}_{J}(\Theta)], \tag{39}\]  
so that  
\[W_{A}^{J}(\Theta)=\sum_{K=\{0,\frac{1}{2}\}}\sum_{q,q^{\prime}=-K}^{K}\sqrt{ \frac{2K+1}{J+1}}A_{\vec{k}q}^{\frac{1-q^{\prime}}{2}\frac{1-q^{\prime}}{2}}D _{qq^{\prime}}^{K*}(\Theta). \tag{40}\]  
One immediately finds the reconstruction to be  
\[\hat{A}_{J}=\frac{J+1}{16\pi^{2}}\int dV\,W_{A}^{J}(\Theta)\ \hat{\omega}(\Theta), \tag{41}\]  
where \(dV=\sin\theta d\phi d\theta d\psi\). Note that if \(\hat{A}\) acts in a single irrep of SU(2), then \(q^{\prime}=0\) in this equation and we recover the standard symbol as discussed in the previous section. An important remark is in order. Since we do not fix the dimension of the group representation, the quantizer--and consequently the associated symbols--depends on three angles. In this setting, the mapping defined in (40) does not correspond to a representation of operators on a classical phase space, which must possess even dimensionality.