Generating random density matrices
Résumé
We study various methods to generate ensembles of quantum density matrices of a fixed size N and analyze the corresponding probability distributions P(x), where x denotes the rescaled eigenvalue, x=N\lambda. Taking a random pure state of a two-partite system and performing the partial trace over one subsystem one obtains a mixed state represented by a Wishart--like matrix W=GG^{\dagger}, distributed according to the induced measure and characterized asymptotically, as N -> \infty, by the Marchenko-Pastur distribution. Superposition of k random maximally entangled states leads to another family of explicitly derived distributions, describing singular values of the sum of k independent random unitaries. Taking a larger system composed of 2s particles, constructing $s$ random bi-partite states, performing the measurement into a product of s-1 maximally entangled states and performing the partial trace over the remaining subsystem we arrive at a random state characterized by the Fuss-Catalan distribution of order s. A more general class of ensembles of random states containing also Bures and arcsine ensembles is constructed.