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 evolution described by a minimal set of raising and lowering Lindblad operators. We demonstrate that thermalisation is typically 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 ρ of the randomly evolving state under these dynamics, which may be contrasted with the impurity of the more frequently considered ensemble average of ρ. The system adopts stationary statistics, with zero stochastic entropy production, after a further stage of relaxation once purity has been achieved. We show that apparent pathological mathematical difficulties in the computation of stochastic entropy production emerge in this evolution towards stationarity if ρ is represented using a certain set of coordinates, though these problems can be removed by choosing a different set. We conclude that frameworks for modelling open systems must be carefully selected to provide satisfactory thermodynamic as well as dynamic behaviour.