N^{m} x N^{m}平均密度矩阵的SU（N）x S_{m} -不变特征空间
SU(N) x S_{m}-Invariant Eigenspaces of N^{m} x N^{m} Mean Density Matrices
摘要：
我们扩展到额外的概率措施和情况下，某些最近的结果Krattenthaler和斯莱特（quant-ph/9612043），其最初的动机是获得量子模拟的开创性工作的通用数据压缩的克拉克和巴伦。KS通过对2 × 2密度矩阵的m重张量积求平均，得到了2 × 2密度矩阵的特征值和特征向量的显式表达式.加权是相对于一个单参数（u）的概率分布族进行的，其中所有成员在二能级量子系统的“布洛赫球”上都是球对称的（SU（2）不变）。对于u = 1/2，获得最小单调（Bures）度量的归一化体积元素。在本文中，平行于KS的分析进行，根据另一种“自然”的措施，最近提出的密度矩阵Zyczkowski，Horodecki，Sanpera，和Lewenstein（quant-ph/9804024）。KS方法和基于ZHSL的方法得到了[1 + m/2]个相同的SU（2）x S_{m}不变特征空间（但当m > 3时特征值不重合）.基于ZHSL测度的SU（3）形式，得到了3 × 3密度矩阵的二重和三重张量积的相关结果.我们发现一个相当显着的限制程序（选择规则）从这些分析中恢复，（置换对称）SU（3）的多重态由两个或三个粒子构成。我们还分析了N = 2 x 3，N= 2 x 3 x 2，N= 3 x 2 x 2和N = 4的情况（所有m = 2），此外，推广了ZHSL测度，使其包含一个（对称）Dirichlet分布族（而不仅仅是均匀分布），定义在特征值的（N-1）维单纯形上。
英文摘要：
We extend to additional probability measures and scenarios, certain of the recent results of Krattenthaler and Slater (quant-ph/9612043), whose original motivation was to obtain quantum analogs of seminal work on universal data compression of Clarke and Barron. KS obtained explicit formulas for the eigenvalues and eigenvectors of the 2^m x 2^m density matrices derived by averaging the m-fold tensor products with themselves of the 2 x 2 density matrices. The weighting was done with respect to a one-parameter (u) family of probability distributions, all the members of which are spherically-symmetric (SU(2)-invariant) over the ``Bloch sphere'' of two-level quantum systems. For u = 1/2, one obtains the normalized volume element of the minimal monotone (Bures) metric. In this paper, analyses parallel to those of KS are conducted, based on an alternative "natural" measure on the density matrices recently proposed by Zyczkowski, Horodecki, Sanpera, and Lewenstein (quant-ph/9804024). The approaches of KS and that based on ZHSL are found to yield [1 + m/2] identical SU(2) x S_{m}-invariant eigenspaces (but not coincident eigenvalues for m > 3). Companion results, based on the SU(3) form of the ZHSL measure, are obtained for the twofold and threefold tensor products of the 3 x 3 density matrices. We find a rather remarkable limiting procedure (selection rule) for recovering from these analyses, the (permutationally-symmetrized) multiplets of SU(3) constructed from two or three particles. We also analyze the scenarios (all for m = 2) N = 2 x 3, N= 2 x 3 x 2, N= 3 x 2 x 2 and N = 4 and, in addition, generalize the ZHSL measure, so that it incorporates a family of (symmetric) Dirichlet distributions (rather than simply the uniform distribution), defined on the (N-1)-dimensional simplex of eigenvalues.
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