Equilibration of apparent temperature in non-equilibrium steady state of Lindblad dynamics

Temperature in a non-equilibrium system is not well defined. If the non-equilibrium dynamics is Lindbladian, it is possible to associate several apparent temperatures with it. However, the apparent temperature governed by the Lindblad equation and its properties as the system approaches a steady state have not been extensively studied. Representing a finite-dimensional quantum system as a graph, we extended Kirchhoff's matrix-tree theorem to a case where the system's graph is not strongly connected and showed that the diagonal and off-diagonal components of a density matrix can be dynamically evolved separately. We discovered that the apparent temperatures of different interaction channels between the system's graph and the environment can be different. It can be equilibrated in each channel if the system has only cycles with length ℓ≤2. In contrast, when the system has a cycle with length ℓ>2, the apparent temperatures of the system and environment are different at a steady state or not equilibrated. Equilibration of an apparent temperature may occur in some channels but not in others and can be undefined in some cases. Generally, an apparent temperature is not transitive because it depends on the interaction channels between the subsystem and its environment. Considering the apparent temperature for each interaction channel, instead of one value of temperature representing the entire system, will provide more insights about the state of a quantum system and potentially lead to more efficient control of quantum systems for many applications in quantum technology. We demonstrated these points in some important finite-dimensional quantum systems pertaining to quantum memory, thermalizing channels in a qutrit and quantum sensing.