  Semiconductor quantum light sources are favorable for a wide range of quantum
photonic tasks, particularly quantum computing and quantum information
processing. Here we theoretically investigate the properties of quantum
emitters (QEs) as a source of entangled photons with practical quantum
properties including heralding of on-demand single photons. Through the
theoretical analysis, we characterize the properties of a cascade (biexciton)
emitter, including (1) studies of single-photon purity, (2) investigating the
first- and second- order correlation functions, and (3) determining the Schmidt
number of the entangled photons. The analytical expression derived for the
Schmidt number of the cascade emitters reveals a strong dependence on the ratio
of decay rates of the first and second photons. Looking into the joint spectral
density of the generated biphotons, we show how the purity and the degree of
entanglement are connected to the production of heralded single photons.
  Our model is further developed to include polarization effects, fine
structure splitting, and the emission delay between the exciton and biexciton
emission. The extended model offers more details about the underlying mechanism
of entangled photon production, and it provides additional degrees of freedom
for manipulating the system and characterizing purity of the output photon. The
theoretical investigations and the analysis provide a cornerstone for the
experimental design and engineering of on-demand single photons.
