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We study the problem of reconstructing an unknown matrix M, of rank r and dimension d, using O(rd poly log d) Pauli measurements. This has applications to compressed sensing methods for quantum state tomography.  We give a solution to this problem based on the restricted isometry property (RIP), which improves on previous results using dual certificates. In particular, we show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r RIP. This implies that M can be recovered from a fixed ("universal") set of Pauli measurements, using nuclear-norm minimization (e.g., the matrix Lasso), with nearly-optimal bounds on the error. Our proof uses Dudley's inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.

The covariant formulation of loop quantum gravity has developed strongly during the last few years.  I summarize the current definition of the theory and the results that have been proven. I discuss what is missing towards of the goal of defining a consistent quantum theory whose classical limit is general relativity.