# Phase Space Insights: Wigner Functions for Qubits and Beyond
## 3 A Menagerie of Wigner Functions
Several theoretical proposals [92; 93; 94; 95; 96; 97; 98] have aimed to surpass this limit, reducing phase variance to the Heisenberg limit, \(\operatorname{Var}\Phi\sim 1/N^{2}\). However, these methods rely on two key constraints: they require the phase difference to be either zero or very small to achieve optimal scaling, and they assume access to the asymptotic limit of a large number of measurement repetitions. For limited resources, it is better to look for states that give optimal phase precision for single-shot estimation with no prior information. These states were identified by [PERSON] and [PERSON] in ref. [99]; they have the expression:  
\[|\psi_{\text{BW}}\rangle=\frac{1}{\sqrt{S+1}}\sum_{m=-S}^{S}\sin\left[\frac{(S +m+1)\pi}{2S+2}\right]|S,m\rangle. \tag{28}\]  
They are particularly effective when combined with adaptive measurement strategies, where phase shifts are dynamically adjusted based on prior outcomes, allowing for precise estimation over a broad range of phase values. The Wigner function of Berry-Wiseman states must be computed numerically, with the results displayed in Figure 2. As observed, the central peak undergoes increased squeezing as the parameter \(S\) grows, accompanied by small side oscillations. Figure 1: Density plots on the unit Bloch sphere of the Wigner functions for spin coherent states located at the \(X\) axis, with the indicated values of \(S\). The scale of the colormap is indicated at the right. \[W_{\text{e}}=\sqrt{\frac{4\pi}{2S+1}}\sum_{K=0}^{2S}\sum_{q=-K}^{K}\varrho_{Kq }\,Y_{Kq}(\theta,\phi). \tag{23}\]0 Density plots on the unit Bloch sphere of the Wigner functions for the optimal Berry–Wiseman states (28) with the indicated values of \(S\). The scale of the colormap is indicated at the right. In the same vein, spin-squeezed states have been proposed to enhance the precision of phase estimation beyond the SQL [100]. However, their advantage is primarily realized in the asymptotic limit, where multiple measurement repetitions allow for statistical averaging. In contrast, for single-shot measurements or scenarios where the phase shift is not necessarily small or well-known, spin-squeezed states may not be the optimal choice. Consider the case where \(S\) is integer. Then, a state which comes close to minimizing the squeezing parameter is the Yurke state [101]:  
\[W_{\text{e}}=\sqrt{\frac{4\pi}{2S+1}}\sum_{K=0}^{2S}\sum_{q=-K}^{K}\varrho_{Kq }\,Y_{Kq}(\theta,\phi). \tag{23}\]1  
where \(|S,m\rangle_{y}\) denote eigenstates of \(\hat{S}_{y}\). The minimum value of the squeezing parameter (which is of the order \(1/S\)) is achieved as \(\alpha\to 0\)[102]. In this limit, the state is invariant under a rotation of \(\pi\) around the \(z\) axis. That is, in a single-shot measurement, it would be impossible to distinguish between a rotation of \(\varphi\) and one of \(\varphi+\pi\). Yurke states exhibit quantum correlations and interference properties that make them particularly well-suited for interferometric phase estimation. These states enable sensitivity beyond the standard quantum limit by leveraging entanglement and quantum superposition. Their Wigner functions, depicted in Figure 3, provide a visual representation of their phase space structure, revealing a high degree of symmetry. This symmetry is a key indicator of their potential for phase estimation, as it reflects the underlying coherence and phase resolution capabilities of the state. Finally, we consider the famous NOON states [103], which, when expressed in the \(|S,m\rangle\) basis, take the form  
\[W_{\text{e}}=\sqrt{\frac{4\pi}{2S+1}}\sum_{K=0}^{2S}\sum_{q=-K}^{K}\varrho_{Kq }\,Y_{Kq}(\theta,\phi). \tag{23}\]2  
Like Yurke states, NOON states enable phase measurements with a sensitivity \(\sqrt{S}\) times better than that of a coherent state [100]. Their key advantage lies in their ability to achieve Heisenberg-limited phase estimation. However, they are invariant under a \(z\)-rotation of \(\pi/S\), which imposes a strict requirement on prior knowledge of the phase.