  Learning physical properties of high-dimensional states is crucial for
developing quantum technologies but usually consumes an exceedingly large
number of samples which are difficult to afford in practice. In this Letter, we
use the methodology of quantum metrology to tackle this difficulty, proposing a
strategy built upon entangled measurements for dramatically reducing sample
complexity. The strategy, whose characteristic feature is symmetrization of
observables, is powered by the exploration of symmetric structures of states
which are ubiquitous in physics. It is provably optimal under some natural
assumption, efficiently implementable in a variety of contexts, and capable of
being incorporated into existing methods as a basic building block. We apply
the strategy to different scenarios motivated by experiments, demonstrating
exponential reductions in sample complexity.
