The power of random measurements: measuring Tr(\rho^n) on single copies   of \rho
  While it is known that Tr(\rho^n) can be measured directly (i.e., without first reconstructing the density matrix) by performing joint measurements on n copies of the same state rho, it is shown here that random measurements on single copies suffice, too. Averaging over the random measurements directly yields estimates of Tr(\rho^n), even when it is not known what measurements were actually performed (so that one cannot reconstruct \rho). 
S. J. van Enk and C. W. J. Beenakker
quant-ph