The Gibbs-Boltzmann entropy effectively characterizes systems with a significant reliance on initial conditions. Nonetheless, the majority of materials generally exhibit behavior that is independent of their initial conditions. Conversely, Tsallis entropy—a nonextensive entropy that underpins nonextensive statistical mechanics—provides a comprehensive framework for modeling systems that engage with their environment. In this study, we derive the Tsallis formulation of the Clausius inequality in a quantum context and establish an upper limit for the work that can be extracted from a small system within the framework of nonextensive quantum statistical mechanics. Furthermore, by utilizing a principle from quantum information theory—which posits that the erasure of a single bit of information corresponds to the dissipation of a specific quantity of energy into the environment—we formulate an inequality for the erasure process in nonextensive systems. This principle corresponds with the physical law that entropy can be transformed into heat. Additionally, we derive mutual information and integrate quantum feedback control to enhance Maxwell’s demon. Ultimately, we extend the second law of thermodynamics to information processes through the application of Tsallis entropy.
The Gibbs-Boltzmann entropy effectively characterizes systems with a significant reliance on initial conditions. Nonetheless, the majority of materials generally exhibit behavior that is independent of their initial conditions. Conversely, Tsallis entropy—a nonextensive entropy that underpins nonextensive statistical mechanics—provides a comprehensive framework for modeling systems that engage with their environment. In this study, we derive the Tsallis formulation of the Clausius inequality in a quantum context and establish an upper limit for the work that can be extracted from a small system within the framework of nonextensive quantum statistical mechanics. Furthermore, by utilizing a principle from quantum information theory—which posits that the erasure of a single bit of information corresponds to the dissipation of a specific quantity of energy into the environment—we formulate an inequality for the erasure process in nonextensive systems. This principle corresponds with the physical law that entropy can be transformed into heat. Additionally, we derive mutual information and integrate quantum feedback control to enhance Maxwell’s demon. Ultimately, we extend the second law of thermodynamics to information processes through the application of Tsallis entropy.
Gibbs-Boltzmann entropy leads to systems that have a strong dependence on initial conditions. In reality, most materials behave quite independently of initial conditions. Nonextensive entropy or Tsallis entropy leads to nonextensive statistical mechanics. In this paper, we calculate the Tsallis form of Clausius inequality and then determine the upper bound for extracting work from the small system in the nonextensive statistical mechanics with mutual information. In the following, we extract mutual information and adjust Maxwell's demon with quantum feedback control.
We review a general theory of thermodynamics of information processing. The background of this topic is the recently-developed nonequilibrium statistical mechanics and quantum (and classical) information theory. These theories are closely related to the modern technologies to manipulate and observe small systems; for example, macromolecules and colloidal particles in the classical regime, and quantum-optical systems and quantum dots in the quantum regime. First, we review a generalization of the second law of thermodynamics to the situations in which small thermodynamic systems are subject to quantum feedback control. The generalized second law is expressed in terms of an inequality that includes the term of information obtained by the measurement, as well as the thermodynamic quantities such as the free energy. This inequality leads to the fundamental upper bound of the work that can be extracted by a “Maxwell's demon”, which can be regarded as a feedback controller with a memory that stores measurement outcomes. Second, we review generalizations of the second law of thermodynamics to the measurement and information erasure processes of the memory of the demon that is a quantum system. The generalized second laws consist of inequalities that identify the lower bounds of the energy costs that are needed for the measurement and the information erasure. The inequality for the erasure leads to the celebrated Landauer's principle for a special case. Moreover, these inequalities enable us to reconcile Maxwell's demon with the second law of thermodynamics. In these inequalities, thermodynamic quantities and information contents are treated on an equal footing. In fact, the inequalities are model-independent, so that they can be applied to a broad class of information processing. Therefore, these inequalities can be called the second law of “information thermodynamics”.
The study of feedback control inspired by Maxwell's demon has yielded fundamental insights into the relationship between thermodynamics and information. However, its fully quantum formulation, where the controller holds quantum side information that cannot be treated as a classical record, remains incomplete. In this paper, we establish fundamental no-go limits for thermodynamic state conversion while allowing quantum side information to be fed back to the system coherently. From two basic operational principles respected by all physically admissible feedback-control schemes, we derive the tightest possible bounds on the single-shot work of formation and extractable work for arbitrary quantum systems conditioned on arbitrary quantum side information. In the asymptotic limit, these bounds yield a generalized second law of thermodynamics with quantum feedback, governed by a conditional free energy. We further show that this generalized second law is consistent with the traditional second law once the acquisition, processing, and reset of side information are accounted for. These results extend information thermodynamics to the fully quantum regime and provide precise thermodynamic meanings for the negativity of single-shot conditional entropies.
Adiabatic measurements, followed by feedback and erasure protocols, have often been considered as a model to embody Maxwell’s Demon paradox and to study the interplay between thermodynamics and information processing. Such studies have led to the conclusion, now widely accepted in the community, that Maxwell’s Demon and the second law of thermodynamics can peacefully coexist because any gain provided by the demon must be offset by the cost of performing the measurement and resetting the demon’s memory to its initial state. Statements of this kind are collectively referred to as second laws of information thermodynamics and have recently been extended to include quantum theoretical scenarios. However, previous studies in this direction have made several assumptions, particularly about the feedback process and the demon’s memory readout, and thus arrived at statements that are not universally applicable and whose range of validity is not clear. In this work, we fill this gap by precisely characterizing the full range of quantum feedback control and erasure protocols that are overall consistent with the second law of thermodynamics. This leads us to conclude that the second law of information thermodynamics is indeed universal : it must hold for any quantum feedback control and erasure protocol, regardless of the measurement process involved, as long as the protocol is overall compatible with thermodynamics. Our comprehensive analysis not only encompasses new scenarios but also retrieves previous ones, doing so with fewer assumptions. This simplification contributes to a clearer understanding of the theory.
We now turn to the subject of thermodynamics, and to describing the process of thermalization of quantum systems. A quantum system that interacts with another system large enough to obey the usual rules of thermodynamics, is referred to as an open system. The thermal system with which it interacts is often referred to as a thermal bath , or simply as the environment. We will use these terms interchangeably. Modeling the behavior of open quantum systems has many applications. One of these is to model the noise that they are subjected to by their surroundings, noise that we will wish to control using measurement and feedback loops in Chapter 5. There are also two important connections between quantum measurements, information theory, and thermodynamics. The first of these is that the thermodynamic entropy of a macroscopic system can be identified with the information-theoretic entropy that characterizes the lack of knowledge of an observer about the microstates of the system. In turn, if an observer is allowed to make microscopic measurements on the system, then this modifies the thermodynamic rules regarding the conversion of heat to work. Such is the subject of Maxwell’s demon. Curiously it took more than a century to understand how Maxwell’s demon, essentially a microscopic measurement and feedback process, fits within the second law of thermodynamics. In another twist, it turns out that the process of erasing information requires an increase in the thermodynamic entropy of the environment, and this increase is directly related to the amount of information erased.
The first law of thermodynamics, which governs energy conservation, is traditionally formulated as an equality. Surprisingly, we demonstrate that the first law alone implies a universal Landauer-like inequality linking changes in system entropy and energy. However, contrasting with the Landauer principle derived from the second law of thermodynamics, our obtained Landauer-like inequality solely relies on system information and is applicable in scenarios where implementing the Landauer principle becomes challenging. Furthermore, the Landauer-like inequality can complement the Landauer principle by establishing a dual {\it upper} bound on heat dissipation. We illustrate the practical utility of the Landauer-like inequality in dissipative quantum state preparation and quantum information erasure applications. Our findings offer new insights into identifying thermodynamic constraints relevant to the fields of quantum thermodynamics and the energetics of quantum information processing and more specifically, this approach could facilitate investigations into systems coupled to non-thermal baths or scenarios where access to bath information is limited.