  The irreversible motion of an open quantum system can be represented through
an ensemble of state vectors following a stochastic dynamics with piecewise
deterministic paths. It is shown that this representation leads to a natural
definition of the rate of quantum entropy production. The entropy production
rate is expressed in terms of the von Neumann entropy and of the numbers of
quantum jumps corresponding to the various decay channels of the open system.
The proof of the positivity and of the convexity of the entropy production rate
is given. Monte Carlo simulations of the stochastic dynamics of a driven qubit
and of a $\Lambda$-configuration involving a dark state are performed in order
to illustrate the general theory.
