  Modelling the evolution of a system using stochastic dynamics typically
implies a greater subjective uncertainty in the adopted system coordinates as
time progresses, and stochastic entropy production has been developed as a
measure of this change. In some situations the evolution of stochastic entropy
production can be described using an It\^o process, but mathematical
difficulties can emerge if diffusion in the system phase space is restricted to
a subspace of lower dimension. This can arise if there are constants of the
motion, for example, or more generally when there are functions of the
coordinates that evolve without noise. We discuss such a case for an open
three-level quantum system modelled within a framework of Markovian quantum
state diffusion and show how the problem of computing the stochastic entropy
production in such a situation can be overcome. We go on to illustrate how a
nonequilibrium stationary state of the three-level system, with a constant mean
production rate of stochastic entropy, can be established under suitable
environmental couplings.
