In this talk, I discuss a technique for measuring nonlinear functionals of a many-body density matrix, such as Renyi entropies with direct connection to entanglement, without measuring and reconstructing the whole density matrix (i.e. without performing full quantum state tomography). The approach, which has direct connection to Random Matrix Theory and quantum chaos, consists in implementing an ensemble of random unitary evolution operators, applying them on the measured many-body state and extracting the desired functions from ensemble averaged observables . Investigating the generation of such random unitary evolution operators and the scaling of errors in possible experiments, I show that our approach is readily implementable with current technology and widely applicable, in particular in systems where full state tomography is not available. Concretely, I present applications in one and two-dimensional Fermi (Bose-) Hubbard models and lattice Spin models as realized by Rydberg atoms or trapped Ions.
 S. J. van Enk and C. W. J. Beenakker, Phys. Rev. Lett. 108, 110503 (2012).