Recursive Analytical Technique for the Quantum Analysis of a Clauser-Horne-Shimony-Holt Parameter Reduction

An analytic technique for the investigation of the combined impacts of multiple photon pairs and a non-negligible joint spectral density on the measurement of a Clauser-Horne-Shimony-Holt (CHSH) parameter is introduced. We demonstrate this technique for an asymmetric Gaussian-normal joint spectral density of spontaneous parametric down-converted photons using two different techniques for generating polarization entanglement. We find that the correlation of the joint spectral density can improve the CHSH parameter for certain ranges of the squeezing parameter when the polarization-entangled state is generated using a beam-splitter technique. Additionally, we find that the impacts due to the correlation of the joint spectral density are made worse when asymmetry is present in the joint spectral density.