  We derive the spectral density of the equiprobable mixture of two random
density matrices of a two-level quantum system. We also work out the spectral
density of mixture under the so-called quantum addition rule. We use the
spectral densities to calculate the average entropy of mixtures of random
density matrices, and show that the average entropy of the
arithmetic-mean-state of $n$ qubit density matrices randomly chosen from the
Hilbert-Schmidt ensemble is never decreasing with the number $n$. We also get
the exact value of the average squared fidelity. Some conjectures and open
problems related to von Neumann entropy are also proposed.
