  Pumping a nonlinear crystal by an intense radiation results in the optical
parametric generation of photons in two modes (the signal and the idler). The
quantized electromagnetic field in these modes is described by a
continuous-variable quantum state, which is entangled if the pump is a coherent
state produced by a laser. The signal and the idler modes remain populated by
photons even if the pump becomes incoherent (dephased by a medium, superposed
with a thermal state, or produced by an alternative source such as the
superluminescent diode). However, the incoherent pump does effect the
entanglement and purity of the signal and the idler modes, which is of vital
importance for quantum information applications and interferometry. Here we
develop an approach to infer the signal-idler entanglement and purity for a
general quantum incoherent pump with the given Glauber-Sudarshan function. We
show that the signal-idler entanglement is extremely sensitive to the phase
distribution of the pump and illustrate our findings by physically relevant
examples of the incoherent pump: the noisy coherent state, slightly dephased
and phase-averaged coherent states, the thermal state, and states modulated by
the Kerr medium. The effect of an incoherent pump on the combined quadratures
is discussed as well.
