Estimating the spectrum of a density operator

Abstract

Given N quantum systems prepared according to the same density operator ρ, we propose a measurement on the N-fold system which approximately yields the spectrum of ρ. The projections of the proposed observable decompose the Hilbert space according to the irreducible representations of the permutations on N points, and are labeled by Young frames, whose relative row lengths estimate the eigenvalues of ρin decreasing order. We show convergence of these estimates in the limit N\to\infty, and that the probability for errors decreases exponentially with a rate we compute explicitly.