  In recent years, analysis methods for quantum states based on randomized
measurements have been investigated extensively. Still, in the experimental
implementations these methods were typically used for characterizing strongly
entangled states and not to analyze the different families of multiparticle or
weakly entangled states. In this work, we experimentally prepare various
entangled states with path-polarization hyper-entangled photon pairs, and study
their entanglement properties using the full toolbox of randomized
measurements. First, we successfully characterize the correlations of a series
of GHZ-W mixed states using the second moments of the random outcomes, and
demonstrate the advantages of this method by comparing it with the well-known
three-tangle and squared concurrence. Second, we generate bound entangled
chessboard states of two three-dimensional systems and verify their weak
entanglement with a criterion derived from moments of randomized measurements.
