Verifying energy-time entanglement with irregularly sampled correlations

Verifying entanglement with experimental measurements requires that we take the limitations of experimental techniques into account, while still proving that the data obtained could not have been generated from a classical source. In the energy-time degree of freedom, this challenge is exacerbated because realistic high-resolution frequency measurements are obtained as a function of light passing through arbitrary filters positioned at uneven intervals. In this work, we show how the data gathered from these kinds of measurements can be used to fully certify the degree of energy and timing correlations needed to certify energy-time entanglement without having to make special assumptions about the state or the measurement device. We accomplish this by showing how to construct a continuous-variable probability density from the data that can closely estimate, but never overestimate the correlations (and entanglement) actually present in the system, and note that these methods are applicable to all continuous-variable degrees of freedom (e.g., spatial, field quadratures, etc.). We illustrate the feasibility of these methods using frequency and timing correlations obtainable from photon pairs in spontaneous parametric down-conversion.