# Phase Space Insights: Wigner Functions for Qubits and Beyond
## 5 Examples  
Let us illustrate this new approach with a couple of simple examples. First of all, the \(J\)-symbols of the \(\mathfrak{su}(2)\) algebra have the crystal-clear form  
\[W^{J}_{S_{k}}(\Theta)=\sqrt{\frac{l\left(\frac{l}{2}+1\right)}{n_{k}}}\sum_{M=0,1,\ldots}\delta_{JM}. \tag{42}\]  
Here, the \(\delta\)-functions indicate the admissible values of the index \(J\) and \(n_{k}\) are components of the unitary vector \(\mathbf{n}=(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)\).  
As a second example, we consider the two-mode Bell-like states  
\[|\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|1\rangle_{A}|0\rangle_{B}\pm|0\rangle_ {A}|1\rangle_{B}),\qquad|\Phi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}|0 \rangle_{B}\pm|1\rangle_{A}|1\rangle_{B}), \tag{43}\]  
where \(|1\rangle_{A}|0\rangle_{B}\) indicates one photon in mode \(A\) and vacuum in mode \(B\), and so on. These states have been used to explore the bizarre feature of vacuum-one photon entanglement [106; 107; 108; 109], which has been demonstrated in the laboratory [111].  
In the angular momentum basis, they are represented as  
\[|\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|\tfrac{1}{2},\tfrac{1}{2}\rangle\pm| \tfrac{1}{2},-\tfrac{1}{2}\rangle),\qquad|\Phi^{\pm}\rangle=\frac{1}{\sqrt{2}} (|0,0\rangle\pm|1,0\rangle). \tag{44}\]  
Interestingly, each state \(|\Psi^{\pm}\rangle\) belongs to a single irrep with \(S=\tfrac{1}{2}\) and the standard SU(2) Wigner map (5) can be used; the result reads  
\[W_{\Psi^{\pm}}(\theta,\phi)=\frac{1}{2}(1\pm\sqrt{3}\sin\theta\cos\phi). \tag{45}\]  
This Wigner function is plotted in Figure 5. Their complementary character under the swapping of the phase \(\phi\) can be clearly observed.  
In contradistinction, the states \(|\Phi^{\pm}\rangle\) are the superpositions from two irreps with \(S=0\) and \(S=1\), and the generalized Wigner function is required to capture their correlation properties. In particular, the density matrices of these states can be decomposed as  
\[\delta_{\Phi^{\pm}}=\frac{1}{2}\left[\hat{\mathcal{H}}_{00}^{00}\pm\left(\hat{ \mathcal{T}}_{10}^{10}-\hat{\mathcal{T}}_{10}^{01}\right)+\frac{1}{\sqrt{3}} \hat{\mathcal{T}}_{00}^{11}-\sqrt{\tfrac{2}{3}}\hat{\mathcal{T}}_{20}^{11} \right], \tag{46}\]  
Figure 5: Density plots on the unit Bloch sphere of the states \(|\Psi^{\pm}\rangle\). The scale of the colormap is indicated at the right.  
which includes nondiagonal tensors \(\hat{T}_{10}^{10}\) and \(\hat{T}_{10}^{01}\), accounting for the interference between subspaces. A direct calculation gives now  
\[W_{\Phi^{\pm}}^{I}(\Theta)=\frac{1}{2}\left[\delta_{0I}\pm\sqrt{3}\sin\theta \cos\psi\delta_{1I}+\left(\frac{1}{3}-\sqrt{\frac{5}{2}}\frac{3\cos^{2}\theta- 1}{3}\right)\delta_{2J}\right]. \tag{47}\]  
The Wigner function resides in the subspaces corresponding to \(J=0,1,2\). The term \(\delta_{1I}\), however, is associated with the action of the operator \(|0,0\rangle\langle 1,0|+|1,0\rangle\langle 0,0|\), which is nondiagonal in the SU(2) representation basis. This term captures a key characteristic of the Wigner function for Bell-like states; namely, the presence of coherence between different spin polarization states.  
This nondiagonal structure implies that the Wigner function encodes quantum coherence across different irreps, which is reflected in the dependence on the angle \(\psi\). When this term is integrated over \(\psi\), it yields a mixed state \(\sim|0,0\rangle\langle 0,0|+|1,0\rangle\langle 1,0|\), corresponding to \(J=0\) and \(J=2\) terms. It is interesting to observe that the term \(\sim\delta_{1I}\) has a similar structure as the significant contribution in \(W_{\Psi^{\pm}}(\theta,\phi)\), after replacing \(\phi\rightarrow\psi\) in the latter; meanwhile, the phase \(\phi\) typically reflects a relative phase between spin components in a fixed irrep, and the phase \(\psi\) can thus be interpreted as encoding relative phase between different irreps. The pieces \(J=1\) and \(J=2\) are plotted in Figure 6, confirming the previous comments.