  In this work, we study the statistical behavior of entanglement in quantum
bipartite systems under the Hilbert-Schmidt ensemble as assessed by the
standard measure - the von Neumann entropy. Expressions of the first three
exact cumulants of von Neumann entropy are known in the literature. The main
contribution of the present work is the exact formula of the corresponding
fourth cumulant that controls the tail behavior of the distribution. As a key
ingredient in deriving the result, we make use of newly observed unsimplifiable
summation bases that lead to a complete cancellation. In addition to providing
further evidence of the conjectured Gaussian limit of the von Neumann entropy,
the obtained formula also provides an improved finite-size approximation to the
distribution.
