Operational definition of the temperature of a quantum state

TL;DR

This paper introduces an operational notion of temperature for arbitrary quantum states based on their ability to exchange heat with a thermal environment, extending the concept beyond equilibrium states.

Contribution

It defines effective temperatures for quantum states using an operational approach inspired by the Zeroth Law, including scenarios with quantum catalysts and coherences.

Findings

Effective temperatures can be assigned to any quantum state.

Quantum catalysts enable enhanced heat exchange, leading to extreme effective temperatures.

Connections established with existing thermodynamic concepts.

Abstract

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 work, 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.