## Distinguishing generic quantum states

Properties of random mixed states of dimension $N$

distributed uniformly with respect to the Hilbert-Schmidt measure are

investigated. We show that for large $N$, due to the concentration of measure

phenomenon, the trace distance between two random states tends to a fixed number

$1/4+1/\pi$, 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. Asymptotic value for the root fidelity between two random states,