# Phase Space Insights: Wigner Functions for Qubits and Beyond
## 2 Wigner Function for a Spin \(S\)  
Any quantum system with a Hilbert space of finite dimension \(2S+1\) can be formally regarded as a spin \(S\). The dynamical symmetry group of such a system is SU(2), generated by three self-adjoint operators \(\hat{\mathbf{S}}=(\hat{S}_{x},\hat{S}_{x},\hat{S}_{z})^{\top}\) (the superscript \(\top\) denoting the transpose), that satisfy the commutation relations (with \(\hbar=1\) throughout)  
\[[\hat{S}_{x},\hat{S}_{y}]=i\hat{S}_{z}, \tag{1}\]  
and cyclic permutations. We consider here a pure spin-\(S\) state \(|\psi\rangle\) living in the Hilbert space \(\mathcal{H}_{S}\), spanned by the angular momentum basis \(\{|S,m\rangle\ |\ m=-S,\ldots,S\}\) of the simultaneous eigenstates of \(\hat{\mathbf{S}}^{2}\) and \(\hat{S}_{z}\). This space serves as the carrier of the irreducible representation (irrep) of spin \(S\) of SU(2), which is isomorphic to \(\mathbb{C}^{2S+1}\). Since physical states are identified up to an overall phase factor (i.e., by rays in \(\mathcal{H}_{S}\)), the manifold of physically distinct states is given by the projective space \(\mathbb{C}\mathbf{P}^{2S}\)[80]. As heralded in the Introduction, the Jordan-Schwinger representation [64; 65] provides a useful way to realize the operators \(\hat{\mathbf{S}}\) in terms of two independent bosonic modes, specified by operators \(\hat{a}_{1}\) (\(\hat{a}_{1}^{\dagger}\)) and \(\hat{a}_{2}\) (\(\hat{a}_{2}^{\dagger}\)) with standard commutation relations \([\hat{a}_{\lambda},\hat{a}_{\lambda^{\prime}}^{\dagger}]=\delta_{\lambda \lambda^{\prime}}\), with \(\lambda,\lambda^{\prime}\in\{1,2\}\). In this representation, the spin operators read  
\[\hat{S}_{x}=\tfrac{1}{2}(\hat{a}_{1}\hat{a}_{2}^{\dagger}+\hat{a}_{1}^{\dagger }\hat{a}_{2}),\qquad\hat{S}_{y}=\tfrac{i}{2}(\hat{a}_{1}\hat{a}_{2}^{\dagger} -\hat{a}_{1}^{\dagger}\hat{a}_{2}),\qquad\hat{S}_{z}=\tfrac{1}{2}(\hat{a}_{1} ^{\dagger}\hat{a}_{1}-\hat{a}_{2}^{\dagger}\hat{a}_{2}), \tag{2}\]  
which are nothing but the Stokes operators for the two-mode problem [81] and the corresponding phase space is naturally identified with the unit sphere \(\mathcal{S}_{2}\). If we define the operator \(\hat{N}=\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}\), one can verify that  
\[\hat{\mathbf{S}}^{2}=\frac{\hat{N}}{2}\bigg{(}\frac{\hat{N}}{2}+\mathbf{1} \bigg{)}, \tag{3}\]  
where \(\mathbf{1}\) is the identity in \(\mathcal{H}_{S}\). Therefore, the spin \(S\) corresponds to (half) the total number of excitations, whereas the eigenvalue \(m\) corresponds to \(m=(n_{1}-n_{2})/2\), where \(n_{\lambda}\) is the number of excitations in mode \(\lambda\). Furthermore, since \([\hat{N},\hat{\mathbf{S}}]=0\), different irreps can be treated independently. According to the axiomatic approach, we associate each operator \(\hat{A}\) acting on \(\mathcal{H}_{S}\) with its symbol, \(W_{A}(\theta,\phi)\), a \(c\)-number function defined in the unit sphere \((\theta,\phi)\in\mathcal{S}_{2}\). Such a (invertible) map \(\hat{A}\mapsto W_{A}(\theta,\phi)\) depends on the ordering rules of functions of noncommutative operators. Here, we will deal exclusively with the symmetric operator ordering, also known as the Weyl ordering. This ordering is particularly advantageous as it ensures a better semi-classical limit, preserving the connection between quantum and classical descriptions. Additionally, it results in balanced quantum corrections in perturbative expansions and is well-suited for spin systems [82]. For a more thorough and detailed exploration, the reader is encouraged to consult the monograph [83].