  The Bures-Hall distance metric between quantum states is a unique measure
that satisfies various useful properties for quantum information processing. In
this work, we study the statistical behavior of quantum entanglement over the
Bures-Hall ensemble as measured by von Neumann entropy. The average von Neumann
entropy over such an ensemble has been recently obtained, whereas the main
result of this work is an explicit expression of the corresponding variance
that specifies the fluctuation around its average. The starting point of the
calculations is the connection between correlation functions of the Bures-Hall
ensemble and these of the Cauchy-Laguerre ensemble. The derived variance
formula, together with the known mean formula, leads to a simple but accurate
Gaussian approximation to the distribution of von Neumann entropy of
finite-size systems. This Gaussian approximation is also conjectured to be the
limiting distribution for large dimensional systems.
