Certification of high-dimensional entanglement and Einstein-Podolsky-Rosen steering with cold atomic quantum memory

The advent of structured, high-dimensional entangled states brings new possibilities for quantum imaging, information processing and quantum key distribution. We experimentally generate and characterize a spatially entangled state stored in a quantum memory system, using entropic EPR-steering inequality, yielding genuine violation of $1.06\pm0.15$ bits and certifying entanglement of formation of at least $0.70\pm0.15$ ebits. The supremacy of the entropic witness is demonstrated for a wide class of experimentally available states, giving prospects for EPR-steering applications in noisy systems or with lossy quantum channels.