Energetic Footprints of Irreversibility in the Quantum Regime

The extension of thermodynamic laws to the quantum regime has been a significant area of research. While maximal work extraction and minimal work cost have been studied extensively for reversible processes, the limitations imposed by irreversibility remain less explored. Classical thermodynamics links irreversibility, quantified by entropy production, to two energetic consequences: reduced extractable work and surplus heat dissipation. Recent work has shown that quantum systems exhibit an additional form of irreversibility linked to decoherence in the energy basis. This paper aims to clarify the energetic footprints of this quantum irreversibility, examining how it affects both heat exchange and work extraction. The study's importance lies in its potential to optimize controlled quantum operations at low temperatures, where irreversibility plays a crucial role. Many previous studies focused on the ideal, reversible limit of quantum processes. This work takes a different approach by explicitly incorporating irreversibility and examining its consequences on the work extraction and heat exchange. This is important because real-world quantum systems inevitably experience irreversible processes due to interactions with their environment. Understanding the impact of this irreversibility is critical to designing and optimizing quantum technologies.

The paper reviews existing literature on quantum thermodynamics, highlighting studies on maximal work extraction and the role of quantum coherence and correlations in thermodynamic processes. It mentions investigations into the irreversibility of quantum processes using stochastic thermodynamics, the concept of fluctuating quantum entropy production, and experimental measurements of entropy production in mesoscopic quantum systems. The authors acknowledge prior work showing that quantum entropy production in a system interacting with another system (not a bath) involves an additional information flow term. They reference studies that explored the optimal limit of reversible processes in quantum thermodynamics, focusing on unitary and quasi-static evolutions. However, these studies largely neglected the impact of irreversibility on work extraction. The literature review also points out that the link between quantum entropy production and its energetic footprints remained opaque before this study.

The researchers employ an imperfect protocol for work extraction from quantum coherences, extending a previously established optimal protocol to include irreversible steps that are unavoidable in experimental settings. The key methodological element is the use of quantum trajectories, specifically eigenstate trajectory unraveling, to analyze the open system dynamics. This approach enables the construction of distributions for classical and quantum heat exchanges. The authors focus on a specific protocol for extracting work from a quantum system's coherences. This protocol involves quenching of the system Hamiltonian in discrete steps. The protocol is modified to include unavoidable experimental imperfections, resulting in classical and quantum irreversibility. Eigenstate trajectory unraveling is used to identify distributions of classical and quantum heat, allowing for a detailed investigation of thermodynamic quantities beyond macroscopic expectation values. The thermalization process is described by a quantum channel, and minimal and augmented trajectories are constructed. The augmented trajectories aid in separating the decoherence and classical thermalization processes. Stochastic quantum trajectories are employed to analyze the heat fluctuations. These trajectories split into decoherence and classical thermalization trajectories. The stochastic quantum entropy production is derived, showing its relation to the irreversibility of the process. Classical and quantum heat distributions are analyzed, with expressions for stochastic classical and quantum heat provided. The energetic footprints of classical and quantum irreversibility are discussed, comparing the classical case with the quantum case. The paper analyzes the variance in quantum heat, linking it to entanglement generation. The effects of Hamiltonian-covariant quantum channels on entropy production and quantum heat variance are studied, comparing the results for qubits and higher-dimensional systems. The authors verify the work footprint of entropy production by analyzing the stochastic work extracted along each trajectory of the protocol. The average extracted work is then derived and expressed in terms of entropy production and free energy changes. The work footprint of irreversibility is presented for the qubit model, showing how the extracted work decreases with increasing quantum coherence and classical non-thermality. Finally, the authors offer a discussion of how to improve the approach regarding using experimentally measured trajectories rather than eigenstate trajectories.

The authors' key finding is the demonstration that the energetic footprints of irreversibility in the quantum regime differ significantly from the classical case. In the classical regime, the average entropy production is directly linked to the surplus of dissipated heat (Clausius equality). In contrast, in the quantum regime, the average quantum entropy production is not directly linked to the average quantum heat, which is always zero. Instead, the average quantum entropy production is linked to the variance in quantum heat. For qubits, a strong monotonic relationship exists between average quantum entropy production and variance in quantum heat. This relationship is shown to be co-monotonic with the system's energy coherence. Both quantities monotonically decrease under Hamiltonian-covariant channels (combinations of dephasing and depolarization). This strong monotonic relationship breaks down for systems with a larger Hilbert space. Another crucial finding is that both classical and quantum entropy production reduce the extractable work in an equal manner (symmetrically). However, the physical mechanisms underlying this reduction are distinct for classical and quantum irreversibility. Classical irreversibility leads to the dissipation of heat and compensates for non-recoverable work, while quantum decoherence produces no average energy change; the reduction in work arises solely from the increase in system entropy. The study also reveals that the heat footprint of quantum irreversibility does not scale with temperature, but with the system's energy gap – reflecting its quantum nature. Importantly, the study provides a detailed analysis of heat distributions in both classical and quantum regimes, including the variance of classical and quantum heat, and their relationship with the corresponding entropy production. These findings are graphically illustrated using histograms and plots of relevant parameters.

The findings address the research question by detailing the energetic footprints of irreversibility in the quantum regime, showing significant differences from classical thermodynamics. The results challenge the direct analogy between classical and quantum heat dissipation, highlighting the distinct role of quantum coherence in influencing thermodynamic processes. The significance of the results lies in their implications for the optimization of quantum control protocols. The trade-off between classical non-thermality and quantum coherence must be considered when optimizing work extraction or other quantum operations in the presence of decoherence and noise. These results are crucial for developing a comprehensive framework for quantum thermodynamics that explicitly accounts for irreversibility, contributing significantly to the ongoing development of quantum technologies. The authors discuss the implications for work optimization strategies, emphasizing the need to consider the trade-off between classical non-thermality and quantum coherence. The study also contributes to the development of a general quantum thermodynamics framework incorporating irreversibility and aids in assessing the energetic cost of quantum control protocols for enhancing computational and communication performance in noisy environments.

This paper provides a comprehensive analysis of irreversibility's energetic footprints in quantum thermodynamics. The study reveals significant differences between classical and quantum regimes, emphasizing the unique role of quantum coherence. The findings highlight the importance of considering both entropic and energetic factors when optimizing quantum operations. Future research could explore the use of experimentally measured trajectories instead of eigenstate trajectories and address the challenge of defining unique fluctuations in quantum heat for degenerate quantum states.

The study uses eigenstate trajectories, an idealized scenario assuming knowledge of the time-local density operators and the ability to measure in the correct eigenbases. Real-world measurements might not precisely follow these trajectories. Another limitation is the focus on a specific work extraction protocol, limiting the generalizability of some findings to other quantum processes. The analysis of the variance in quantum heat does not uniquely account for states with degenerate eigenvalues, necessitating further research to establish a unique measure in such cases.