Appears in collection : 2017 - T3 - WS2 - Probabilistic techniques and quantum information theory

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/π, 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. We consider also random quantum channels, especially the limiting behaviour of the diamond norm for two independent random quantum operations. In the case of flat measure on random channels, the limiting value of the diamond norm is equal to 1/2+2/π.