  We investigate the spectral statistics of the difference of two density
matrices, each of which is independently obtained by partially tracing a random
bipartite pure quantum state. We first show how a closed-form expression for
the exact joint eigenvalue probability density function for arbitrary
dimensions can be obtained from the joint probability density function of the
diagonal elements of the difference matrix, which is straightforward to
compute. Subsequently, we use standard results from free probability theory to
derive a relatively simple analytic expression for the asymptotic eigenvalue
density (AED) of the difference matrix ensemble, and using Carlson's theorem,
we obtain an expression for its absolute moments. These results allow us to
quantify the typical asymptotic distance between the two random mixed states
using various distance measures; in particular, we obtain the almost sure
asymptotic behavior of the operator norm distance and the trace distance.
