Hall normalization constants for the Bures volumes of the n-state quantum systems

We report the results of certain integrations of quantum-theoretic interest, relying, in this regard, upon recently developed parametrizations of Boya et al (1998 Preprint quant-ph/9810084) of the n × n density matrices, in terms of squared components of the unit (n - 1)-sphere and the n × n unitary matrices. Firstly, we express the normalized volume elements of the Bures (minimal monotone) metric for n = 2 and 3, thereby obtaining `Bures prior probability distributions' over the two- and three-state systems. Then, as a first step in extending these results to n>3, we determine that the `Hall normalization constant' (Cn) for the marginal Bures prior probablity distribution over the (n - 1)-dimensional simplex of the n eigenvalues of the n × n density matrices is, for n = 4, equal to 71 680/2. Since we also find that C3 = 35/, it follows that C4 is simply equal to 211C3/. (C2 itself is known to equal 2/.) The constant C5 is also found. It too is associated with a remarkably simple decomposition, involving the product of the eight consecutive prime numbers from 3 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.