  Properties of random mixed states of order $N$ distributed uniformly with
respect to the Hilbert-Schmidt measure are investigated. We show that for large
$N$, due to the concentration of measure, the trace distance between two random
states tends to a fixed number ${\tilde D}=1/4+1/\pi$, which yields the
Helstrom bound on their distinguishability. To arrive at this result we apply
free random calculus and derive the symmetrized Marchenko--Pastur distribution,
which is shown to describe numerical data for the model of two coupled quantum
kicked tops. Asymptotic values for the fidelity, Bures and transmission
distances between two random states are obtained. Analogous results for quantum
relative entropy and Chernoff quantity provide other bounds on the
distinguishablity of both states in a multiple measurement setup due to the
quantum Sanov theorem.
