Apparent pathologies in stochastic entropy production in the thermalisation of an open two-level quantum system

We investigate the entropic consequences of the relaxation of an open two-level quantum system towards a thermalised statistical state, using a framework of quantum state diffusion with a minimal set of raising and lowering Lindblad operators. We demonstrate that thermalisation starting from a general state is accompanied by a persistent non-zero mean rate of change of the environmental component of stochastic entropy production. This thermodynamic signature can be associated with the purification of the reduced density matrix $\rho$ of the randomly evolving state, to be contrasted with the impurity of the more often considered ensemble average of $\rho$. The system adopts stationary statistics, with zero stochastic entropy production, once purity has been achieved. However, we show that apparent pathological mathematical difficulties in the computation of stochastic entropy production emerge if $\rho$ is represented using a certain set of coordinates, though these can be removed by choosing a different set. We conclude that frameworks for modelling open quantum systems must be carefully selected to provide satisfactory thermodynamic as well as dynamic behaviour.