  Hilbert-Schmidt distance is one of the prominent distance measures in quantum
information theory which finds applications in diverse problems, such as
construction of entanglement witnesses, quantum algorithms in machine learning,
and quantum state tomography. In this work, we calculate exact and compact
results for the mean square Hilbert-Schmidt distance between a random density
matrix and a fixed density matrix, and also between two random density
matrices. In the course of derivation, we also obtain corresponding exact
results for the distance between a Wishart matrix and a fixed Hermitian matrix,
and two Wishart matrices. We verify all our analytical results using Monte
Carlo simulations. Finally, we apply our results to investigate the
Hilbert-Schmidt distance between reduced density matrices generated using
coupled kicked tops.
