  Entanglement is the powerful and enigmatic resource central to quantum
information processing, which promises capabilities in computing, simulation,
secure communication, and metrology beyond what is possible for classical
devices. Exactly quantifying the entanglement of an unknown system requires
completely determining its quantum state, a task which demands an intractable
number of measurements even for modestly-sized systems. Here we demonstrate a
method for rigorously quantifying high-dimensional entanglement from extremely
limited data. We improve an entropic, quantitative entanglement witness to
operate directly on compressed experimental data acquired via an adaptive,
multilevel sampling procedure. Only $6,456$ measurements are needed to certify
an entanglement-of-formation of $7.11 \pm .04$ ebits shared by two
spatially-entangled photons. With a Hilbert space exceeding 68 billion
dimensions, we need $20$-million-times fewer measurements than the uncompressed
approach and $10^{18}$-times fewer measurements than tomography. Our technique
offers a universal method for quantifying entanglement in any large quantum
system shared by two parties.
