  In spontaneous parametric down conversion (SPDC) based quantum information
processing (QIP) experiments, there is a tradeoff between the coincide count
rates (i.e. the pumping power of the SPDC), which limits the rate of the
protocol, and the visibility of the quantum interference, which limits the
quality of the protocol. This tradeoff is mainly caused by the multi-photon
pair emissions from the SPDCs. In theory, the problem is how to model the
experiments without truncating these multi-photon emissions while including
practical imperfections.
  In this paper, we establish a method to theoretically simulate SPDC based
QIPs which fully incorporates the effect of multi-photon emissions and various
practical imperfections. The key ingredient in our method is the application of
the characteristic function formalism which has been used in continuous
variable QIPs. We apply our method to three examples, the Hong-Ou-Mandel
interference and the Einstein-Podolsky-Rosen interference experiments, and the
concatenated entanglement swapping protocol. For the first two examples, we
show that our theoretical results quantitatively agree with the recent
experimental results. Also we provide the closed expressions for these the
interference visibilities with the full multi-photon components and various
imperfections. For the last example, we provide the general theoretical form of
the concatenated entanglement swapping protocol in our method and show the
numerical results up to 5 concatenations. Our method requires only a small
computation resource (few minutes by a commercially available computer) which
was not possible by the previous theoretical approach. Our method will have
applications in a wide range of SPDC based QIP protocols with high accuracy and
a reasonable computation resource.
