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 $Q_{ρ}$ be the Husimi function of a density operator $ρ$ on $% \mathcal{P}_{N}$. We prove an isospectral version of Lieb-Solovej inequality: if $ρ^{\downarrow }$ is obtained by placing the eigenvalues of $ρ$ 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 $Φ$ on $[0,1]$. Applying the corresponding reversed inequality to the concave function $Φ(t)=-t\log t$ gives the Wehrl entropy. In the process it is show that the output state of $ρ$ under Lieb-Solovej's channel is majorized by the output of the state $ρ^{\downarrow }$. 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 $1_Ω$. 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 $T_Ω$, obtained without using the spherical isoperimetric inequality.