Kurtosis of von Neumann entanglement entropy

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