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The convex set of quantum states of a composite $K \times K$ system withpositive partial transpose is analysed. A version of the hit and run algorithmis used to generate a sequence of random points covering this set uniformly andan estimation for the convergence speed of the algorithm is derived. For $K\ge3$ this algorithm works faster than sampling over the entire set of states andverifying whether the partial transpose is positive. The level density of thePPT states is shown to differ from the Marchenko-Pastur distribution, supportedin [0,4] and corresponding asymptotically to the entire set of quantum states.Based on the shifted semi--circle law, describing asymptotic level density ofpartially transposed states, and on the level density for the Gaussian unitaryensemble with constraints for the spectrum we find an explicit form of theprobability distribution supported in [0,3], which describes well the leveldensity obtained numerically for PPT states.

Convex set of quantum states with positive partial transpose analysed by hit and run algorithm