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

Abstract

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 $% \mathcal{P}_{N}$. 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}}\Phi (Q_{\rho }(z))\,dm(z)\leq \int_{\mathbb{C}}\Phi (Q_{\rho ^{\downarrow }}(z))\,dm(z) \end{equation*}% for every convex function $\Phi $ on $[0,1]$. Applying the corresponding reversed inequality to the concave function $\Phi (t)=-t\log t$ 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 $1_{\Omega}$. Equivalently, caps maximize all Ky Fan norms, leading to the sharp inequalities, previously known for $r=1$: \begin{equation*} \sum_{j=1}^{r}\lambda_{j}(\Omega) \leq r-\sum_{k=0}^{r-1}(r-k)\binom{N+1}{k} m(\Omega)^{k}\bigl(1-m(\Omega)\bigr)^{N+1-k}. \end{equation*} This implies isoperimetric inequalities for all Schatten sums of $T_{\Omega}$, obtained without using the spherical isoperimetric inequality.