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The resources needed to conventionally characterize a quantum system areoverwhelmingly large for high- dimensional systems. This obstacle may beovercome by abandoning traditional cornerstones of quantum measurement, such asgeneral quantum states, strong projective measurement, and assumption-freecharacterization. Following this reasoning, we demonstrate an efficienttechnique for characterizing high-dimensional, spatial entanglement with oneset of measurements. We recover sharp distributions with local, randomfiltering of the same ensemble in momentum followed by position---something theuncertainty principle forbids for projective measurements. Exploiting theexpectation that entangled signals are highly correlated, we use fewer than5,000 measurements to characterize a 65, 536-dimensional state. Finally, we useentropic inequalities to witness entanglement without a density matrix. Ourmethod represents the sea change unfolding in quantum measurement where methodsinfluenced by the information theory and signal-processing communities replaceunscalable, brute-force techniques---a progression previously followed byclassical sensing.

Compressively characterizing high-dimensional entangled states with complementary, random filtering