n态量子系统的Bures模型的Hall归一化常数
Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems
摘要：
我们报告的结果，量子理论的兴趣，依赖于最近开发的参数化的Boya等人的n × n密度矩阵，在平方分量的单位（n-1）球和n × n酉矩阵的某些集成。首先，我们表示归一化体积元素的Bures（最小单调）度量n = 2和3，从而获得“Bures先验概率分布”的两个和三个状态的系统。然后，作为将这些结果推广到n > 3的必要的第一步，我们确定在n × n密度矩阵的n个特征值的（n-1）维单纯形上的边缘布勒斯先验概率分布的“霍尔归一化常数”（C_{n}），对于n = 4，等于71680/pi^2。由于我们还发现C_{3} = 35/pi，因此得出C_{4}简单地等于2^{11} C_{3} /pi。（已知C_{2}本身等于2/π。）还发现了常数C_{5}，它也与一个非常简单的分解有关，包括从2到23的8个连续素数的乘积。我们还初步研究了几种情况下，n > 5，使用拟蒙特卡罗积分。我们希望，报告的各种分析将证明在推导任意n的霍尔归一化常数的一般公式（证据表明将涉及伯努利数）时是有用的。这将有不同的应用，包括量子推理和通用量子编码。
英文摘要：
We report the results of certain integrations of quantum-theoretic interest, relying, in this regard, upon recently developed parameterizations of Boya et al of the n x n density matrices, in terms of squared components of the unit (n-1)-sphere and the n x n unitary matrices. Firstly, we express the normalized volume elements of the Bures (minimal monotone) metric for n = 2 and 3, obtaining thereby "Bures prior probability distributions" over the two- and three-state systems. Then, as an essential first step in extending these results to n > 3, we determine that the "Hall normalization constant" (C_{n}) for the marginal Bures prior probability distribution over the (n-1)-dimensional simplex of the n eigenvalues of the n x n density matrices is, for n = 4, equal to 71680/pi^2. Since we also find that C_{3} = 35/pi, it follows that C_{4} is simply equal to 2^{11} C_{3}/pi. (C_{2} itself is known to equal 2/pi.) The constant C_{5} is also found. It too is associated with a remarkably simple decompositon, involving the product of the eight consecutive prime numbers from 2 to 23. We also preliminarily investigate several cases, n > 5, with the use of quasi-Monte Carlo integration. We hope that the various analyses reported will prove useful in deriving a general formula (which evidence suggests will involve the Bernoulli numbers) for the Hall normalization constant for arbitrary n. This would have diverse applications, including quantum inference and universal quantum coding.
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