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

Abstract

The advent of complex, structured, and high-dimensional entangled states brings both new possibilities for experimental and theoretical scenarios and new challenges for the generation and characterization of such states. In particular, spatially structured photonic states offer applications in quantum imaging, information processing, and quantum key distribution. Here we experimentally generate a spatially entangled high-dimensional state composed of at least 10 Schmidt modes in a quantum memory setup and perform characterization using the entropic Einstein-Podolsky-Rosen (EPR)-steering inequality, yielding a genuine violation of $1.06\ifmmode\pm\else\textpm\fi{}0.15$ bits. The entanglement of formation of at least $0.70\ifmmode\pm\else\textpm\fi{}0.15$ ebit for the measured noisy state is certified using the entropic witness method. We point out and solve the difficulties in estimating the entropy, achieving characterization of the high-dimensional entangled state with highly undersampled data. Finally, the practical supremacy of the entropic EPR-steering witness over a variance-based witness is demonstrated for a wide class of states typical in an experimental scenario, giving prospects for EPR-steering demonstrations and applications in noisy systems or with lossy quantum channels.