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 quantph/9810084) of the n x n density matrices, in te rms 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, thereby obtaining 'Bures prior pr obability 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' (C-n) for the marginal Bures prior probablity dist ribution over the (n - 1)-dimensional simplex of the II eigenvalues of the it x n density matrices is, for n = 4, equal to 71 680/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 t oo is associated with a remarkably simple decomposition, involving the prod uct of the eight consecutive prime numbers from 3 to 23. We also preliminar ily investigate several cases n > 5, with the use of quasi-Monte Carlo inte gration. We hope that the Various analyses reported will prove useful in de riving a general formula (which evidence suggests will involve the Bernoull i numbers) for the Hall normalization constant for arbitrary n. This would have diverse applications, including quantum inference and universal quantu m coding.