  A prominent tool to study the dynamics of open quantum systems is the reduced
density matrix. Yet, approaching open quantum systems by means of state vectors
has well known computational advantages. In this respect, the physical meaning
of the so-called conditional states in Markovian and non-Markovian scenarios
has been a topic of recent debate in the construction of stochastic
Schr\"{o}dinger equations. We shed light on this discussion by acknowledging
the Bohmian conditional wavefunction as the proper mathematical object to
represent, in terms of state vectors, an arbitrary subset of degrees of
freedom. As an example of the practical utility of these states, we present a
time-dependent quantum Monte Carlo algorithm to describe electron transport in
open quantum systems under general (Markovian or non-Markovian) conditions. By
making the most of trajectory-based and wavefunction methods, the resulting
simulation technique extends, to the quantum regime, the computational
capabilities that the Monte Carlo solution of the Boltzmann transport equation
offers for semi-classical electron devices.
