Distinguishability of 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, the trace distance between two random states tends to a fixed number $\widetilde{D}=1/4+1/\pi$, 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 coupled quantum kicked tops. Asymptotic value for the root fidelity between two random states, $\sqrt{F} = \frac{3}{4}$, can serve as a universal reference value for further theoretical and experimental studies. 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. We study also mean entropy of coherence of random pure and mixed states and entanglement of a generic mixed state of a bipartite system