functions, we find that $$\label{Euler}
(\alpha \partial_\alpha + \beta \partial_\beta)V(\alpha,\beta) = 2 J V(\alpha,\beta)$$ with an identical equation for $\alpha^*$ and $\beta^*$.

Relation between the two approaches
-----------------------------------

Equation (\[Q\]) can be related to the spin coherent state function in Eq. (\[spin\_Q\]) as follows. Define the angle-action variables $\mathcal{I}$, $\theta$, $\phi$, and $\psi$ according to $$\label{action_angles}
\begin{aligned}
\alpha &= \sqrt{\mathcal{I}} \cos\frac{\theta}{2} e^{-i (\phi+\psi)/2}, \qquad\beta = \sqrt{\mathcal{I}} \sin\frac{\theta}{2} e^{i (\phi-\psi)/2}.
\end{aligned}$$ The integration measure changes as $${\,\mathrm{d}}^2\alpha {\,\mathrm{d}}^2 \beta = \frac{1}{8} \mathcal{I} {\,\mathrm{d}}\mathcal{I}  {\,\mathrm{d}}\psi {\,\mathrm{d}}\Omega.$$ After integrating over $\psi$, we obtain $$\label{eq:vol_element}
{\,\mathrm{d}}^2\alpha {\,\mathrm{d}}^2 \beta = \frac{\pi}{4} \; \mathcal{I} {\,\mathrm{d}}\mathcal{I} {\,\mathrm{d}}\Omega,$$ where ${\,\mathrm{d}}\Omega = \sin\theta {\,\mathrm{d}}\theta {\,\mathrm{d}}\phi$. One may then verify that the Husimi functions (\[Q\]) and (\[spin\_Q\]) are related by $$\label{relation_Qs}
Q(\alpha,\beta) = \frac{e^{-\mathcal{I}} \mathcal{I}^{2J}}{\pi^2 (2J)!} \; \mathcal{Q}(\Omega).$$ Thus, one may move back and forth between the two representations based on convenience. Comparing this result with Eq. (\[Q\]) also allows us to identify the relation $$V(\alpha,\beta) = \frac{\mathcal{I}^{2J}}{(2J)!} \mathcal{Q}(\Omega).$$

Wehrl entropy
-------------

The entropy associated to the Husimi function is known as the Wehrl entropy [@Wehrl1978; @Wehrl1979; @Anderson1993; @General1995; @Zyczkowski2001; @DelReal2013a; @Gnutzmann2001], $$\label{entropy_spin}
S = - \frac{(2J+1)}{4\pi} \int {\,\mathrm{d}}\Omega \; \mathcal{Q}(\Omega) \ln \mathcal{Q}(\Omega),$$ where the constant $(2J+1)/4\pi$ has been introduced 