Operational Definition of the Temperature of a Quantum State

Temperature is usually defined for physical systems at thermal equilibrium. Nevertheless one may wonder if it would be possible to attribute a meaningful notion of temperature to an arbitrary quantum state, beyond simply the thermal (Gibbs) state. In this Letter, we propose such a notion of temperature considering an operational task, inspired by the zeroth law of thermodynamics. Specifically, we define two effective temperatures for quantifying the ability of a quantum system to cool down or heat up a thermal environment. In this way we can associate an operationally meaningful notion of temperature to any quantum density matrix. We provide general expressions for these effective temperatures, for both single- and many-copy systems, establishing connections to concepts previously discussed in the literature. Finally, we consider a more sophisticated scenario where the heat exchange between the system and the thermal environment is assisted by a quantum reference frame. This leads to an effect of "coherent quantum catalysis," where the use of a coherent catalyst allows for exploiting quantum energetic coherences in the system, now leading to much colder or hotter effective temperatures. We demonstrate our findings using a two-level atom coupled to a single mode of the electromagnetic field.