Estimating the spectrum of a density operator
Michael Keyl, Reinhard F. Werner · Physical Review A · 2001
Given N quantum systems prepared according to the same density operator $\ensuremath{\rho},$ we propose a measurement on the N-fold system that approximately yields the spectrum of $\ensuremath{\rho}.$ 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 $\ensuremath{\rho}$ in decreasing order. We show convergence of these estimates in the limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ and that the probability for errors decreases exponentially with a rate we compute explicitly.