AI 中文总结
该研究针对布洛赫球上的SU(2)相干态，证明Lieb-Solovej不等式的等谱版本，应用得到Toeplitz算子相关等周不等式，未用球面等周不等式。
AI 中文摘要
设$\boldsymbol{\textit{P}}_{N}$为次数至多为$N$的解析多项式构成的$(N+1)$维希尔伯特空间，这是定义$SU(2)$（布洛赫）相干态的自然环境。设$\boldsymbol{Q}_{\rho}$为$\boldsymbol{\textit{P}}_{N}$上密度算子$\boldsymbol{\rho}$的胡西米函数。我们证明了Lieb-Solovej不等式的等谱版本：若$\boldsymbol{\rho}^{\boldsymbol{\text{↓}}}$是通过将$\boldsymbol{\rho}$的特征值按单项式基降序排列得到的，则对$[0,1]$上的任意凸函数$\boldsymbol{\textit{\textit{\textbf{Φ}}}}$，有$\boldsymbol{\textit{\textbf{∫}}}_{\boldsymbol{\textbf{C}}}\boldsymbol{\textit{\textbf{Φ}}}(\boldsymbol{Q}_{\boldsymbol{\rho}}(\boldsymbol{z}))\boldsymbol{\textit{\textbf{d}}}\boldsymbol{\textit{\textbf{m}}}(\boldsymbol{z})\boldsymbol{\textit{\textbf{≤}}}\boldsymbol{\textit{\textbf{∫}}}_{\boldsymbol{\textbf{C}}}\boldsymbol{\textit{\textbf{Φ}}}(\boldsymbol{Q}_{\boldsymbol{\rho}^{\boldsymbol{\text{↓}}}}(\boldsymbol{z}))\boldsymbol{\textit{\textbf{d}}}\boldsymbol{\textit{\textbf{m}}}(\boldsymbol{z})$。将对应的反向不等式应用于凹函数$\boldsymbol{\textit{\textbf{Φ}}}(\boldsymbol{t})\boldsymbol{=}\boldsymbol{-}\boldsymbol{t}\boldsymbol{\text{log}}\boldsymbol{t}$可得到Wehrl熵。在此过程中，证明了在Lieb-Solovej信道下$\boldsymbol{\rho}$的输出态被$\boldsymbol{\rho}^{\boldsymbol{\text{↓}}}$的输出态优化。作为应用，在球面所有面积固定的可测子集$\boldsymbol{\textit{Ω}}$中，球形帽最大化了符号为$\boldsymbol{1}_{\boldsymbol{\textit{Ω}}}$的Toeplitz算子特征值的所有部分和；等价地，球形帽最大化了所有Ky Fan范数，得到了此前仅对$\boldsymbol{r}\boldsymbol{=}\boldsymbol{1}$已知的严格不等式：$\boldsymbol{\textit{\textbf{∑}}}_{\boldsymbol{j}\boldsymbol{=}\boldsymbol{1}}^{\boldsymbol{r}}\boldsymbol{\textit{\textbf{λ}}}_{\boldsymbol{j}}(\boldsymbol{\textit{Ω}})\boldsymbol{\textit{\textbf{≤}}}\boldsymbol{r}\boldsymbol{-}\boldsymbol{\textit{\textbf{∑}}}_{\boldsymbol{k}\boldsymbol{=}\boldsymbol{0}}^{\boldsymbol{r}\boldsymbol{-}\boldsymbol{1}}(\boldsymbol{r}\boldsymbol{-}\boldsymbol{k})\boldsymbol{\textit{\textbf{C}}}_{\boldsymbol{N}\boldsymbol{+}\boldsymbol{1}}^{\boldsymbol{k}}\boldsymbol{\textit{\textbf{m}}}(\boldsymbol{\textit{Ω}})^{\boldsymbol{k}}\boldsymbol{(}\boldsymbol{1}\boldsymbol{-}\boldsymbol{\textit{\textbf{m}}}(\boldsymbol{\textit{Ω}})\boldsymbol{)}^{\boldsymbol{N}\boldsymbol{+}\boldsymbol{1}\boldsymbol{-}\boldsymbol{k}}$。这意味着$\boldsymbol{T}_{\boldsymbol{\textit{Ω}}}$的所有Schatten和的等周不等式，且未使用球面等周不等式。
英文摘要
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.
Comments25 pages, 1 figure
