Estimating the spectrum of a density operator - art. no. 052311

Given N quantum systems prepared according to the same density operator rho . we propose a measurement on the N-fold system that approximately yields t he spectrum of rho. The projections of the proposed observable decompose th e Hilbert space according to the irreducible representations of the permuta tions on N points. and are labeled by Young frames, whose relative row leng ths estimate the eigenvalues of rho in decreasing order. We show convergenc e of these estimates in the limit N --> proportional to, and that the proba bility for errors decreases exponentially with a rate we compute explicitly .