Local temperatures and voltages in quantum systems far from equilibrium

We show that the local measurement of temperature and voltage for a quantum system in steady state, arbitrarily far from equilibrium, with arbitrary interactions within the system, is unique when it exists. This is interpreted as a consequence of the second law of thermodynamics. We further derive a necessary and sufficient condition for the existence of a solution. In this regard, we find that a solution occurs whenever there is no net population inversion. However, when there is a net population inversion, we may characterize the system with a (unique) negative temperature. These results provide a firm mathematical foundation for our measurement protocol, and sound meaning to such measurements in the thermodynamic sense.