发表机构
School of Mathematical Sciences, Hangzhou Dianzi University（杭州电子科技大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究固定二分密度矩阵的谱并随机本征基，通过HCIZ积分等方法推导不同子系统维度下边缘态谱的概率分布公式，验证了密度的分段多项式性质。
AI 中文摘要
固定二分密度矩阵的谱并按Haar测度随机其本征基，研究由此诱导出的两个边缘态谱的概率分布。对任意子系统维数m和n，约化密度矩阵的联合特征函数可表示为Harish-Chandra-Itzykson-Zuber（HCIZ）积分，其外本征值为两两和x_i + y_j；重复外本征值通过合流行列式极限处理。在两量子比特情形，推导得到两个边缘Bloch半径联合密度的显式交替样条公式，其支撑域为Bravyi-Klyachko兼容区域；还得到单个量子比特边缘Bloch半径密度的紧凑截断幂公式。在量子比特-qutrit情形，推导得到量子比特Bloch半径密度的截断幂公式，以及qutrit边缘最大与最小本征值的联合二元样条公式——因迹固定，后两个变量确定qutrit的完整谱。推导结合了合流HCIZ积分、分布傅里叶逆变换、轨道测度及SU(2)和SU(3)导数原理；所得密度在由固定全局本征值的子集和确定的腔室上为分段多项式，与投影余伴随轨道测度的Duistermaat-Heckman描述一致。
英文摘要
Fix the spectrum of a bipartite density matrix and randomize its eigenbasis according to Haar measure. We study the probability distributions induced on the spectra of the two marginal states. For arbitrary subsystem dimensions $m$ and $n$, the joint characteristic function of the reduced density matrices is expressed as a Harish-Chandra-Itzykson-Zuber integral whose external eigenvalues are the pairwise sums $x_i + y_j$. Repeated external eigenvalues are handled by confluent determinant limits. In the two-qubit case, we derive an explicit alternating-spline formula for the joint density of the two marginal Bloch radii. Its support is the Bravyi-Klyachko compatibility region. We also obtain a compact truncated-power formula for the Bloch-radius density of either individual qubit marginal. In the qubit-qutrit case, we derive a truncated-power formula for the qubit Bloch-radius density and a bivariate spline formula for the joint density of the largest and smallest eigenvalues of the qutrit marginal. The latter two variables determine the full qutrit spectrum because the trace is fixed. The derivations combine confluent HCIZ integrals, distributional Fourier inversion, orbital measures, and the SU(2) and SU(3) derivative principles. The resulting densities are piece-wise polynomial on chambers determined by subset sums of the fixed global eigenvalues, in agreement with the Duistermaat-Heckman description of projected coadjoint-orbit measures.
Comments45 pages, 5 figures
