Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere

Let ${\mathcal{P}}_N$ be the $(N+1)$-dimensional Hilbert space of analytic polynomials of degree at most $N$. This is the natural environment to define $SU(2)$ (Bloch) coherent states. Let $Q_{\rho }$ be the Husimi function of a density operator $\rho$ on $$. We prove an isospectral version of Lieb-Solovej inequality: if ${\rho }^{\downarrow }$ is obtained by placing the eigenvalues of $\rho$ in decreasing order along the monomial basis, then \begin{equation*} \int_{$\mathbb{C}$}Φ(Q_{ρ}(z)),dm(z)\leq \int_{$\mathbb{C}$}Φ(Q_{ρ^{\downarrow }}(z)),dm(z) \end{equation*}% for every convex function $\mathrm{\Phi }$ on $[0,1]$. Applying the corresponding reversed inequality to the concave function $\mathrm{\Phi }(t)=-t\log t$ gives the Wehrl entropy. In the process it is show that the output state of $\rho$ under Lieb-Solovej’s channel is majorized by the output of the state ${\rho }^{\downarrow }$. As an application, among all measurable subsets of the sphere having a fixed area, spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol $1_{\mathrm{\Omega }}$. Equivalently, caps maximize all Ky Fan norms, leading to the sharp inequalities, previously known for $r=1$: \begin{equation*} \sum_{j=1}^{r}λ_{j}(Ω) \leq r-\sum_{k=0}^{r-1}(r-k)\binom{N+1}{k} m(Ω)^{k}\bigl(1-m(Ω)\bigr)^{N+1-k}. \end{equation*} This implies isoperimetric inequalities for all Schatten sums of $T_{\mathrm{\Omega }}$, obtained without using the spherical isoperimetric inequality.