Out-of-equilibrium quantum thermodynamics in the Bloch sphere: temperature and internal entropy production

An explicit expression for the temperature of an open two-level quantum system is obtained as a function of local properties, under the hypothesis of weak interaction with the environment. This temperature is defined for both equilibrium and out-of-equilibrium states, and coincides with the environment temperature if the system reaches thermal equilibrium with a heat reservoir. Additionally, we show that within this theoretical framework the total entropy production can be partitioned into two contributions: one due to heat transfer, and another, associated to internal irreversibilities, related to the loss of internal coherence by the qubit. The positiveness of the heat capacity is established, as well as its consistency with the well known results at thermal equilibrium. We apply these concepts to two different systems, and show that they behave in analogous ways as their classical counterparts.