# Phase Space Insights: Wigner Functions for Qubits and Beyond
## 3 A Menagerie of Wigner Functions  
The primary application of SU(2) in metrology is likely phase estimation. To this end, we will attempt to illustrate quantum states commonly employed in quantum interferometry for such tasks and assess their usefulness based on these visualizations. In this endeavor, the Wigner function introduced in this work proves highly valuable, as it enables us to visualize the symmetry of a quantum state with respect to a given axis. We start with SU(2) (or spin) coherent states [89; 90], as they are the most classical states allowed by quantum mechanics. Their robustness against decoherence and their natural emergence in systems with SU(2) symmetry make them a fundamental tool in both theoretical and experimental quantum information. They are defined in direct analogy with the standard coherent states, as a displaced version on the sphere of a fiducial state that we take as the north pole:  
\[\ket{\theta,\phi}=\hat{D}(\theta,\phi)\ket{S,S}, \tag{20}\]  
where the displacement over the sphere is given in terms of the SU(2) ladder operators \(\hat{S}_{\pm}=\hat{S}_{x}\pm\hat{S}_{y}\) by \(\hat{D}(\theta,\phi)=\exp[i\theta/2(e^{-i\phi}\hat{S}_{+}-e^{+i\phi}\hat{S}_{-})]\). They can be expressed as  
\[\ket{\theta,\phi}=\sum_{m=-S}^{S}\left( \begin{array}{c}2S\\ S+m\end{array} \right)^{1/2}\left(\sin\frac{\theta}{2}\right)^{S+m}\left(\cos \frac{\theta}{2}\right)^{S-m}e^{-i(S+m)\phi}\ket{S,m}. \tag{21}\]  
The corresponding multipoles can be readily computed [62]; the result reads  
\[\varrho_{Kq}^{\text{coh}} =e^{i\eta\phi}\left(\tan\frac{\theta}{2}\right)^{q}\sum_{m=-S}^{ S}(-1)^{S-m}\left( \begin{array}{c}2S\\ S+m\end{array} \right)^{1/2}\left( \begin{array}{c}2S\\ S+m+q\end{array} \right)^{1/2}\] \[\times\left(\sin\frac{\theta}{2}\right)^{2(S+m)}\left(\cos\frac{ \theta}{2}\right)^{2(S-m)}C_{Sm,S-m-q}^{K-q}, \tag{22}\]  
and, according to the general recipe in (13), the Wigner function turns out to be  
\[W_{\text{e}}=\sqrt{\frac{4\pi}{2S+1}}\sum_{K=0}^{2S}\sum_{q=-K}^{K}\varrho_{Kq }\,Y_{Kq}(\theta,\phi). \tag{23}\]  
If, for simplicity, we assume that the coherent state is centered along the \(x\) axis (that is \(\ket{\theta=\pi/2,\varphi=0}\), so that it is an eigenstate of \(\hat{S}_{x}\)), we have  
\[\ket{\pi/2,0}=\frac{1}{2^{S}}\sum_{m=-S}^{S}\sqrt{\frac{(2S)!}{(S+m)!(S-m)!}} \ket{S,m}. \tag{24}\]  
From here, we obtain  
\[W_{\text{coh}}(\theta,\phi)=\sqrt{\frac{4\pi}{2S+1}}\sum_{K=0}^{2S}\sqrt{ \frac{2K+1}{2S+1}}\sum_{q=-K}^{K}Y_{Kq}^{*}(\theta,\phi)\sum_{m,m^{\prime}=-S} ^{S}C_{Sm,Kq}^{Sm^{\prime}}\,\alpha_{mm^{\prime}}, \tag{25}\]  
with  
\[\alpha_{mm^{\prime}}=\frac{1}{2^{2S}}\frac{(2S)!}{(S+m)!(S-m)!(S+m^{\prime})! (S-m^{\prime})!}. \tag{26}\]  
This expression is involved and can only be evaluated numerically. However, in the limit \(S\gg 1\), this Wigner function simplifies to an approximate form, enabling more intuitive analytical insights [83]:  
\[W_{\text{coh}}(\theta,\phi)\simeq(\sin\theta\cos\phi)^{2S-1}(1+\sin\theta\cos \phi). \tag{27}\]  
This Wigner function for the coherent state (24) is represented for various values of \(S\) in Figure 1. As we can appreciate, this function is sharply peaked around the classical phase space point \((\vartheta=\pi/2,\varphi=0)\) and spreads over an area proportional to \(1/S\). As \(S\) increases, the state becomes increasingly localized and so more classical. The ultimate constraint on measurement precision in interferometry is dictated by quantum effects. This limitation is most effectively analyzed using a Mach-Zehnder interferometer, where output measurements serve to estimate the relative phase \(\Phi\) between its two arms. It is well established that, when an \(N\)-photon number state is injected into one input port, the achievable phase sensitivity follows the standard quantum limit (SQL) [91], scaling as \(\operatorname{Var}\Phi\sim 1/N\), where \(\operatorname{Var}\) denotes the variance.