n × n密度矩阵上单调矩阵的体积元作为热力学状态密度。II
Volume Elements of Monotone Metrics on the n x n Density Matrices as Densities-of-States for Thermodynamic Purposes. II
摘要：
在n = 3和4的情况下，我们得到了n × n密度矩阵的（n^{2}-1）维凸集上最小和最大单调度量的体积元的显式表达式.我们取得了进一步的进展，为特定的n = 3的最大单调的情况下，通过采取一定比例的积分结果的限制，使用正交的一组八个坐标。通过这样做，我们找到了基于相应体积元素的非常简单的边缘概率分布，然后我们将其用于热力学目的。因此，我们找到了一个自旋为1的类似朗之万函数。然而，在完全一般的n = 4的情况下，由于不能对角化3 × 3的厄米矩阵，从而不能获得用于必要的积分的正交坐标集，我们在取得类似的进展时受到阻碍。
英文摘要：
We derive explicit expressions for the volume elements of both the minimal and maximal monotone metrics over the (n^{2} - 1)-dimensional convex set of n x n density matrices for the cases n = 3 and 4. We make further progress for the specific n = 3 maximal-monotone case, by taking the limit of a certain ratio of integration results, obtained using an orthogonal set of eight coordinates. By doing so, we find remarkably simple marginal probability distributions based on the corresponding volume element, which we then use for thermodynamic purposes. We, thus, find a spin-1 analogue of the Langevin function. In the fully general n = 4 situation, however, we are impeded in making similar progress by the inability to diagonalize a 3 x 3 Hermitian matrix and thereby obtain an orthogonal set of coordinates to use in the requisite integrations.
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