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Irreversible Entropy Production, from Quantum to Classical

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Irreversible Entropy Production, from Quantum to Classical Irreversible entropy production, from quantum to classical Gabriel T. Landi1, Mauro Paternostro2∗ 1Instituto de F´ısica da Universidade de Sao˜ Paulo, 05314-970 Sao˜ Paulo, Brazil 2Centre for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom (Dated: July 7, 2021) Entropy production is a key quantity in any finite-time thermodynamic process. It is intimately tied with the fundamental laws of thermodynamics, embodying a tool to extend thermodynamic consid- erations all the way to non-equilibrium processes. It is also often used in attempts to provide the quantitative characterization of logical and thermodynamic irreversibility, stemming from processes in physics, chemistry and biology. Notwithstanding its fundamental character, a unifying theory of entropy production valid for general processes, both classical and quantum, has not yet been formu- lated. Developments pivoting around the frameworks of stochastic thermodynamics, open quantum systems, and quantum information theory have led to substantial progress in such endeavour. This has culminated in the unlocking of a new generation of experiments able to address stochastic ther- modynamic processes and the impact of entropy production on them. This paper aims to provide a compendium on the current framework for the description, assessment and manipulation of entropy production. We present both formal aspects of its formulation and the implications stemming from the potential quantum nature of a given process, including a detailed survey of recent experiments. CONTENTS F. Pauli master equation for multiple baths 25 G. Classical phase space 25 I. Introduction2 H. Quantum phase space 26 A. Irreversible thermodynamics3 VI. Resource theoretic approach 29 II. Why entropy production matters3 A. Thermo-majorization 30 A. Operation of heat engines3 B. The second laws of thermodynamics 32 B. Heat and particle flow4 C. Coherence and the resource theory of asymmetry 32 C. Landauer’s erasure5 D. Fluctuating work in the resource theory context 33 D. Thermodynamic Uncertainty Relations5 E. Reconciliation with the stochastic approach 35 E. Fluctuation theorems6 F. Stochastic thermodynamics6 VII. Applications 36 G. Maxwell, Szilard and information thermodynamics7 A. The SWAP engine 36 B. Stroke-based engines 38 III. Entropy production in quantum processes7 C. Squeezed baths 39 A. Global unitary dynamics for system + environment7 D. Quantum heat 40 B. Thermal environments8 E. Infinitesimal quenches 41 C. Maps with global fixed points9 F. Dissipative phase transitions: basic models 44 D. Strict energy conservation and thermal operations9 G. Dissipative phase transitions: entropy production 45 E. Fluctuation theorems 10 H. Effects of non-Markovian dynamics on entropy production 47 F. Non-equilibrium lag 12 VIII. Experimental assessment of quantum entropy production 49 arXiv:2009.07668v3 [quant-ph] 5 Jul 2021 IV. Information-theoretic aspects 14 A. Assessment at the level of quantum trajectories 49 A. Corrections to Landauer’s principle 14 B. Assessment of the effects of quantum measurements 50 B. Conditional entropy production 15 C. Assessment of the non-equilibrium Landauer principle 50 C. Heat flow in the presence of correlations 16 D. Assessment of entropy production in non-equilibrium D. Fluctuation theorem under classical and quantum correlations 17 steady-states 52 V. Quantum dynamics and the classical limit 18 IX. Conclusions 54 A. Collisional models 19 Acknowledgments 55 B. The emergence of a preferred basis 20 C. Continuous-time limit 21 References 56 D. On/off work and Spohn’s separation 22 E. Pauli master equations and Schnackenberg’s approach 23 ∗ [email protected] 2 I. INTRODUCTION Sec. I.A, and then discuss – in Sec.II – why the entropy pro- duction problem is relevant. We then move to explore the In every finite-time process, entropy may flow from one consequences and ramifications of such unified formulation. system to another. However, entropy does not satisfy a Sec.IV focuses on information-theoretic corrections to Lan- continuity equation so that it may also be irreversibly pro- dauer principle and the role of classical and quantum corre- duced (Carnot, 1824; Clausius, 1854, 1865). Such entropy lations in heat flow. Sec.V embodies another essential part production, Σ, is always non-negative and zero only in the lim- of the review. We use the concepts developed in Sec. III as iting case where the process is reversible. It therefore serves building blocks to assess the entropy production in more gen- as the key quantity behind the 2nd law of thermodynamics, eral types of dynamics, constructed in terms of a collisional which can be stated mathematically as model. This allows us to address the classical limit as a par- ticular case of the quantum formulation. Σ ≥ 0: (1) The link between information and thermodynamics has Albeit compact, this expression has far reaching conse- other far-reaching consequences, as it allows information to quences, as it places severe restrictions on the types of trans- be cast as a resource, on equal footing to traditional thermo- formations allowed in a physical process. At a foundational dynamic resources, such as heat and work. That is, informa- level, the statement embodied by Eq. (1) manifests the lack of tion can be consumed, stored or interconverted into other re- time-reversal in finite-time processes and stems from the exis- sources. And it can be used to fuel thermodynamic tasks, such tence of an arrow of time (Batalhao˜ et al., 2015; Blythe, 2008; as Maxwell Demon Engines. This is the topic of Sec.VI. Costa et al., 2005; Eddington, 1928; Gaspard, 2004; Jiang In the quantum domain this acquires additional significance et al., 2003; Luo et al., 1984; Mackey, 1989, 1992; Maes and due to the possibility of manipulating quantum coherence, as Netocyny, 2003; Parrondo et al., 2009; Pomeau, 1982; Por- well as quantum correlations such as discord and entangle- porato et al., 2007; Qian, 2001; Schnakenberg, 1976; Seifert, ment. How these features are implemented within a quantum 2005). Hence, the characterization and assessment of irre- formulation of the entropy production problem, is a central versible entropy production is one of the most important tasks theme of this review. in non-equilibrium physics. The formulation of the entropy production problem, how- Finally, Secs. VII and VIII discuss applications and exper- ever, is not universal. It depends on the underlying physi- iments. There is an inevitable arbitrariness on the choice of cal system, as well as its governing dynamical laws. Despite papers to cover, and we have chosen to address those which this, during the last century several widely applicable frame- we believe are (i) representative of the types of problems the works have been developed, from Onsager’s reciprocity the- community is currently interested in; and (ii) have the poten- ory (de Groot and Mazur, 1961; Onsager, 1931a) to the cel- tial to open unexplored avenues of research. Concerning the ebrated fluctuation theorems (Campisi et al., 2011; Esposito experiments, we have also tried to focus on those contribu- et al., 2009; Goold et al., 2016; Vinjanampathy and Anders, tions which specifically characterize the entropy production 2016). More recently, the demonstrated possibility to control at the quantum level. elementary quantum systems has drawn attention to the poten- tial for thermodynamic applications in the quantum domain. We finish this review in Sec.IX, by taking a step back to This is the primary drive towards a formulation of a theory look at the bigger picture. We compare the formulation put of entropy production capable of encompassing both classical forth in Sec. III with other approaches, both historical and and quantum features. modern. The main argument we make is that the 2nd law The goal of this review is to provide an overview of the is always formulated by starting with a basic physical prin- progress in this formulation. Our approach will be cen- ciple, such as those of Carnot, Clausius and Kelvin, or state- tered around a unified picture of the 2nd law, described in ments such as “the entropy of the universe never decreases.” terms of global system-environment quantum unitary inter- One then asks what are the overarching consequences of this actions [cf. Sec. III]. This allows us to establish a link with principle, and which other principles can be derived from it. information theory, and construct entropy production solely This provides a measure of how general it is. The informa- in terms of information-theoretic quantities. The result is a tion theoretic formulation of Sec.IX falls under this category. generalized form of the 2nd law, valid beyond the standard However, its main advantage is that it starts by assuming full paradigms of thermodynamics, but with classical results re- knowledge of all degrees of freedom involved, thus allow- covered in the suitable limit. This approach also has a clear ing for precise mathematical statements. Irreversibility is then operational interpretation, with irreversibility emerging from constructed operationally, by specifying which sources of in- the restrictions on the allowed set of operations for a given formation can, or cannot, be known in a given process. This thermodynamic process. For a broader perspective on the de- feature greatly generalizes the breadth and scope of the 2nd velopments in Quantum Thermodynamics over the last two law. It not only contains
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