  We show that combining randomized measurement protocols with importance
sampling allows for characterizing entanglement in significantly larger quantum
systems and in a more efficient way than in previous work. A drastic reduction
of statistical errors is obtained using classical techniques of
machine-learning and tensor networks using partial information on the quantum
state. In present experimental settings of engineered many-body quantum systems
this effectively doubles the (sub-)system sizes for which entanglement can be
measured. In particular, we show an exponential reduction of the required
number of measurements to estimate the purity of product states and GHZ states.
