  The lossy propagation law (generalization of Lambert-Beer's law for classical
radiation loss) for non-classical, dual-mode entangled states is derived from
first principles, using an infinite-series of beam splitters to model
continuous photon loss. This model is general enough to accommodate
stray-photon noise along the propagation, as well as amplitude attenuation. An
explicit analytical expression for the density matrix as a function of
propagation distance is obtained for completely general input states with
bounded photon number in each mode. The result is analyzed numerically for
various examples of input states. For N00N state input, the loss of coherence
and entanglement is super exponential as predicted by a number of previous
studies. However, for generic input states, where the coefficients are
generated randomly, the decay of coherence is very different; in fact no worse
than the classical Beer-Lambert law. More surprisingly, there is a plateu at a
mid-range interval in propagation distance where the loss is in fact
sub-classical, following which it resumes the classical rate. The qualitative
behavior of the decay of entanglement for two-mode propagation is also analyzed
numerically for ensembles of random states using the behavior of negativity as
a function of propagation distance.
