Objective Quantum thermodynamics is an emerging field that extends the results of classical thermodynamics to the quantum world. The main research topic is based on the axioms of quantum mechanics, and reinterprets the three laws of thermodynamics from the microscopic perspective, with the help of new tools originated from the research of quantum information science. One of the most important results of quantum thermodynamics is the re-characterization of the second law of thermodynamics, which explains and quantifies the state-transformation ability of classical states under the so-called thermodynamic operation (a type of free quantum operations defined in the resource theory of quantum thermodynamics). But there are still some open problems in this field that are hard to solve, such as what role quantum coherence plays in thermodynamics. Coherence, which originates from the phenomenon of quantum superposition, is an indispensable resource in the field of quantum information and computing, and is also one of the essential sources of the difference between the microscopic world and the macroscopic world. In quantum thermodynamics, the coherence is characterized based on the covariance of thermodynamic operations (TOs) and thus has some unique properties. In this review, we reveal some special behaviors of unspeakable coherence in quantum thermodynamics from two aspects according to our previous results.
Methods Thermal operations refer to the quantum operations for the target system after maintaining the energy conservation of the composite system of heat reservoir and main system. Although it has a clear meaning in physics, it is hard to analyze mathematically. Therefore, the previous studies have proposed enhanced thermodynamic operations (EnTOs) satisfying weaker conditions to simplify the mathematical analysis. It is found that the population dynamics of the enhanced thermodynamic operations is equivalent to that of the original thermodynamic operations in any finite dimension of the target system, but the coherence dynamics of the high dimensional system (of which dimension is larger than 3) is actually difficult to examine. It is discovered that there are state conversions under EnTO which cannot be realized exactly by TO. However, it remains an open problem whether this gap can be closed approximately. In order to investigate this problem, in our study, we first proposed a strict subset of thermodynamic operations, single-mode thermodynamic operations, which have the experimental-friendly and equally gapped single-mode bosonic heat reservoir, and then developed a series of results corresponding to its coherence dynamics.
Catalysis is a concept of resource theory, which is described as applying a composite free operation on the target system and the resourceful auxiliary quantum system, and enhancing the power of free operations without disturbing the auxiliary quantum system. In the resource theory of asymmetry or unspeakable coherence, there exists a no-broadcasting theorem, that is, incoherent states cannot obtain coherence resources through catalytic covariant operations, even allows the existence of correlation between target system and catalytic system. Thermodynamic operation is a special type of covariant operation (in different contexts, it also refers to as translational invariant operation or symmetric operation), thus the previous study on no-broadcasting theorem disproved the simple generalization from the incoherent second law of quantum thermodynamics to a full version of the second law. But through a series of constructive protocol, we find that the no-broadcasting theorem is actually unstable, i.e., even the smallest amount of asymmetry in the catalytic system can still strictly enhance the power of a covariant operation.
Results and Discussions In the part of our research on a single-mode thermodynamic operation, first, we show that for single-qubit target systems, the single-mode thermodynamic operation has the same coherence transformation ability as the enhanced thermodynamic operation. But for higher-dimensional systems, such as three-level systems, there is a non-negligible gap in the coherence transformation ability between the single-mode thermodynamic operation and the enhanced thermodynamic operation. This result is a key step in solving the gap conjecture between enhanced thermodynamic operation and thermodynamic operation. At the same time, we also compare the population dynamics of three types of thermodynamic operations (
In part of amplification of asymmetry with correlated catalyst, we first prove that, when the catalytic system is in a pure state, the no-catalyst theorem of asymmetry in any finite dimension is still valid, no matter whether the target system initially has asymmetric resource or not. Second, for the qubit system, we prove that if there is a small amount of coherence in the target system, it can be amplified by the symmetric operation with a correlated catalyst. Furthermore, it follows that the set of symmetric operations with correlated catalyst is almost equivalent to the whole set of quantum operations for qubit under approximate conditions. Finally, we develop a set of numerical methods for the research on correlated catalytic symmetric operations, which can also be generalized to the research on enhanced thermodynamic operations.
Conclusions We reveal the peculiar behaviors of unspeakable coherence in quantum thermodynamics from two aspects. In the first part, our research on coherence dynamics of single-mode thermodynamic operations provides new methods on exploring the gap between thermodynamic operations and enhanced thermodynamic operations, and the fact that erasing correlation is resource-consuming may attract more research interest in the quantum correlation behaviors of thermodynamics. In the second part, since the previous studies on the thermodynamic operations of catalysis have focused less on the coherence behaviors, the results here are of great significance for studying the full version of the second law of quantum thermodynamics under the correlated catalysis condition.
