Hall normalization constants for the Bures volumes of then-state quantum systems
Paul B. Slater · Journal of Physics A Mathematical and General · 1999
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' ( C n ) 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 C 3 = 35/ , it follows that C 4 is simply equal to 2 11 C 3 / . ( C 2 itself is known to equal 2/ .) The constant C 5 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.