Qudit entanglement and steering certification in energy-time domain

Here we demonstrate the certification of high-dimensional entanglement and EPR steering using energy-time measurements. At the qudit source, we certify a lower bound of the maximum quantum state fidelity (ρ,) of 96.2 ± 0.2%, an entanglement-of-formation  of 3.0 ± 0.1 ebits, an entanglement dimensionality d_ent of 24, and a lower bound of steering robustness  of 8.9 ± 0.1 which corresponds to a Schmidt number n certification of a 9-dimensional quantum steering. We sent our qudit resource to dispersion conditions and maintain 21-dimensional entanglement, with the maximum quantum state fidelity (ρ,) of  93.1 0.3%, an  of 2.50.1 ebits. Besides, we witness a 7-dimensional entanglement in a semi-device independent manner in this setting.