  Monte Carlo (MC) simulations are the standard tool for describing jet-like
multi-particle final states. To apply them to the simulation of medium-modified
jets in heavy ion collisions, a probabilistic implementation of medium-induced
quantum interference effects is needed. Here, we analyze in detail how the
quantum interference effects included in the BDMPS-Z formalism of
medium-induced gluon radiation can be implemented in a quantitatively
controlled, local probabilistic parton cascade. The resulting MC algorithm is
formulated in terms of elastic and inelastic mean free paths, and it is by
construction insensitive to the IR and UV divergences of the total elastic and
inelastic cross sections that serve as its basic building blocks in the
incoherent limit. Interference effects are implemented by reweighting gluon
production histories as a function of the number of scattering centers that act
within the gluon formation time. Unlike existing implementations based on gluon
formation time, we find generic arguments for why a quantitative implementation
of quantum interference cannot amount to a mere dead-time requirement for
subsequent gluon production. We validate the proposed MC algorithm by comparing
MC simulations with parametric dependencies and analytical results of the
BDMPS-Z formalism. In particular, we show that the MC algorithm interpolates
correctly between analytically known limiting cases for totally coherent and
incoherent gluon production, and that it accounts quantitatively for the
medium-induced gluon energy distribution and the resulting average parton
energy loss. We also verify that the MC algorithm implements the transverse
momentum broadening of the BDMPS-Z formalism. We finally discuss why the
proposed MC algorithm provides a suitable starting point for going beyond the
approximations of the BDMPS-Z formalism.
