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Learning physical properties of high-dimensional states is crucial fordeveloping quantum technologies but usually consumes an exceedingly largenumber of samples which are difficult to afford in practice. In this Letter, weuse the methodology of quantum metrology to tackle this difficulty, proposing astrategy built upon entangling measurements for dramatically reducing samplecomplexity. The strategy, whose characteristic feature is symmetrization ofobservables, is powered by the exploration of symmetric structures of stateswhich are ubiquitous in physics. It is provably optimal under some naturalassumption, efficiently implementable in a variety of contexts, and capable ofbeing incorporated into existing methods as a basic building block. We applythe strategy to different scenarios motivated by experiments, demonstratingexponential reductions in sample complexity.

Inferring physical properties of symmetric states from the fewest copies