A photon-pair source feeding real detectors lives in an enormous Hilbert space - many time-frequency modes, multipair emission, lost photons and spurious clicks - yet an experimenter usually wants a single two-qubit density matrix that reproduces everything the coincidence counter sees. This concept covers how to derive that effective state from basic probability. Pairs arrive Poisson-distributed with mean mu, each photon is detected with probability eta absorbing both channel and detector efficiency, and each detector fires spuriously with dark-count probability Pd under a Bernoulli model. Writing the probability of a coincidence as a sum over the ground-truth pair assignment (multinomial) and the observed click pattern lets four laboratory configurations be treated as one expression: four photon-number-resolving detectors, four threshold detectors, two PNR, and two threshold, since a threshold detector is the special case of summing over one or more clicks and an absent detector is the sum over all outcomes. The four-PNR case yields an exact effective density matrix valid for any parameters; the other three need low-efficiency, low-flux approximations, whose accuracy is checked numerically against the exact visibility. The physical conclusions are the interesting part: going from two to four detectors appreciably raises fidelity and concurrence because the extra detectors veto events the two-detector setup cannot see, while photon-number resolution buys almost nothing in these regimes, since the dominant multipair noise is created by loss rather than by unresolved multiphoton clicks.
We develop a comprehensive model for the effective two-photon density matrix produced by a parametric source of entangled photon pairs under a variety of detector configurations commonly seen in a la…