In this work we will show that there exists a fundamental difference between microscopic quantum thermodynamics and macroscopic classical thermodynamics. It will be proved that the entropy production in quantum thermodynamics always vanishes for both closed and open quantum thermodynamic systems. This novel and very surprising result is derived based on the genuine reasoning Clausius used to establish the science of thermodynamics in the first place. This result will interestingly lead to define the generalized temperature for any non-equilibrium quantum system.
We compare the thermodynamic entropy of a quantum Brownian oscillator derived from the partition function of the subsystem with the von Neumann entropy of its reduced density matrix. At low temperatures we find deviations between these two entropies which are due to the fact that the Brownian particle and its environment are entangled. We give an explanation for these findings and point out that these deviations become important in cases where statements about the information capacity of the subsystem are associated with thermodynamic properties, as it is the case for the Landauer principle.
We discuss some features of thermodynamics in the presence of multiple conserved quantities. We prove a generalisation of Landauer principle illustrating tradeoffs between the erasure costs paid in different "currencies". We then show how the maximum entropy and complete passivity approaches give different answers in the presence of multiple observables. We discuss how this seems to prevent current resource theories from fully capturing thermodynamic aspects of non-commutativity.
We introduce a generalized approach to characterize the non-Markovianity of quantum dynamical maps via breakdown of monotonicity of thermodynamic functions. By adopting an entropy-based formulation of quantum thermodynamics, we use the relationship between heat and entropy to propose a measure of non-Markovianity based on the heat flow for single-qubit quantum evolutions. This measure can be applied for unital dynamical maps that do not invert the sign of the internal energy. Under certain conditions, it can also be extended for other thermodynamic functions, such as internal energy and work f…
Thermodynamics is usually developed starting from entropy and the maximum entropy principle. We investigate here to what extent one can replace entropy with relative entropy which has several advantages, for example in the context of local quantum field theory. We find that the principle of maximum entropy can be replaced by a principle of minimum expected relative entropy. Various ensembles and their thermodynamic potentials can be defined through relative entropy. We also show that thermal fluctuations are in fact governed by a relative entropy. Furthermore we reformulate the third law of th…
We investigate a total thermodynamic entropy production rate of an isolated quantum system. In particular, we consider a quantum model of coupled harmonic oscillators in a star configuration, where a central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath). In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature. This enables us to define time-dependent thermodynamic entropy f…
It is argued that a typical many body energy eigenstate has a well defined thermodynamic entropy and that individual eigenstates possess thermodynamic characteristics analogous to those of generic isolated systems. We examine large systems with eigenstate energies equivalent to finite temperatures. When quasi-static evolution of a system is adiabatic (in the quantum mechanical sense), two coupled subsystems can transfer heat from one subsystem to another yet remain in an energy eigenstate. To explicitly construct the entropy from the wave function, degrees of freedom are divided into two unequ…
Thermodynamic entropy, as defined by Clausius, characterizes macroscopic observations of a system based on phenomenological quantities such as temperature and heat. In contrast, information-theoretic entropy, introduced by Shannon, is a measure of uncertainty. In this Letter, we connect these two notions of entropy, using an axiomatic framework for thermodynamics [Lieb, Yngvason, Proc. Roy. Soc.(2013)]. In particular, we obtain a direct relation between the Clausius entropy and the Shannon entropy, or its generalisation to quantum systems, the von Neumann entropy. More generally, we find that …
The quantum thermodynamic property of the fractional damping system is investigated extensively. A fractional power-law decaying entropy function is revealed which presents another evidence for the validity of the third law of thermodynamics in the quantum dissipative region. Several non-trivial characters are excavated such as that the entropy varies from a non-linear diverging function to a semi-linear decaying function of the fractional exponent as the temperature tends to absolute zero.
Similarity between quantum mechanics and thermodynamics is discussed. It is found that if the Clausius equality is imposed on the Shannon entropy and the analogue of the heat quantity, then the value of the Shannon entropy comes to formally coincide with that of the von Neumann entropy of the canonical density matrix, and pure-state quantum mechanics apparently transmutes into quantum thermodynamics. The corresponding quantum Carnot cycle of a simple two-state model of a particle confined in a one-dimensional infinite potential well is studied, and its efficiency is shown to be identical to th…
We provide a quantum statistical basis for (a)a characterisation of a complete set of thermodynamic variables and (b) the differentiability of the entropy function of these variables
We extend classical coarse-grained entropy, commonly used in many branches of physics, to the quantum realm. We find two coarse-grainings, one using measurements of local particle numbers and then total energy, and the second using local energy measurements, which lead to an entropy that is defined outside of equilibrium, is in accord with the thermodynamic entropy for equilibrium systems, and reaches the thermodynamic entropy in the long-time limit, even in genuinely isolated quantum systems. This answers the long-standing conceptual problem, as to which entropy is relevant for the formulatio…
Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-Hawking (black hole) entropy, which relates the entropy to the cross-sectional area of the black hole horizon. Using generalized uncertainty principle (GUP), corrections to the geometric entropy and thermodynamics of black hole will be introduced. The impact of GUP on the entropy near the horizon of three types of black holes; Schwarzschild, Garfinkle-Horowitz-Strominger and Reissner-Nordström is determined. It is found that the logarithmic divergence in the entropy-area relation turns to be …
We give meaning to the first and second laws of thermodynamics in case of mesoscopic out-of-equilibrium systems which are driven by diffusion processes. The notion of the entropy production is analyzed. The role of the Helmholtz extremum principle is contrasted to that of the more familiar entropy extremum principles.
The Boltzmann-Gibbs-von Neumann entropy of a large part (of linear size L) of some (much larger) d-dimensional quantum systems follows the so-called area law (as for black holes), i.e., it is proportional to $L^{d-1}$. Here we show, for d=1,2, that the (nonadditive) entropy S_q satisfies, for a special value of $q \neq 1$, the classical thermodynamical prescription for the entropy to be extensive, i.e., $S_q \propto L^d$. Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited, by M. Gell-Mann, Y. Sato and one…
We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as $S_d=-\sum_n ρ_{nn}\ln ρ_{nn}$ with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy $S_n=-{\rm Tr}\, ρ\lnρ$. However, in contrast to $S_n$, the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of thermodynamics the d-entropy can only increase or stay the same. We also show that the d-entropy can be expressed through the energy…
We develop a Landauer-Büttiker theory of entropy evolution in time-dependent strongly coupled electron systems. This formalism naturally avoids the problem of system-bath distinction caused by the strong hybridization of central system and surrounding reservoirs. In an adiabatic expansion up to first order beyond the quasistatic limit, it provides a clear understanding of the connection between heat and entropy currents generated by time-dependent potentials and shows their connection to the occurring dissipation. Combined with the work required to change the potential, the developed formalism…
We combine the formalisms of diagonal entropy and Jarzynski Equality to study the thermodynamic properties of closed quantum systems. Applying this approach to a quantum harmonic oscillator, the diagonal entropy offers a notion of temperature for closed systems away from equilibrium, and allows computing free-energy profiles. We also apply this approach to a hard-core boson lattice model, and discuss measures how to estimate temperature, entropy and measure work distribution functions. This technique offers a path to investigate the non-equilibrium quantum dynamics by means of performing work …
We argue that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics when we excite the system. In relativistic setups, its effective temperature is proportional to the inverse of the subsystem size. This provides a universal relationship between the energy and the amount of quantum information. We derive the results using holography and confirm them in two dimensional field theories. We will also comment on an example with negative specific heat and suggest a connection between the second law of thermodynamics and the strong …
A dispersive medium becomes entangled with zero-point fluctuations in the vacuum. We consider an arbitrary array of material bodies weakly interacting with a quantum field and compute the quantum mutual information between them. It is shown that the mutual information in D dimensions can be mapped to classical thermodynamic entropy in D+1 dimensions. As a specific example, we compute the mutual information both analytically and numerically for a range of separation distances between two bodies in D=2 dimensions and find a logarithmic correction to the area law at short separations. A key advan…
We review recent progress in understanding certain aspects of the thermodynamics of black holes and other horizons. Our discussion centers on various ``entropy bounds'' which have been proposed in the literature and on the current understanding of how such bounds are {\it not} required for the semi-classical consistency of black hole thermodynamics. Instead, consistency under certain extreme circumstances is provided by two effects. The first is simply the exponential enhancement of the rate at which a macrostate with large entropy is emitted in any thermal process. The second is a new sense i…
This thesis investigates the connection between quantum theory, thermodynamics and information theory. Theories with structure similar to that of quantum theory are considered, mathematically described by the framework of "Generalized Probabilistic Theories". For these theories, a thought experiment by von Neumann is adapted to obtain a natural thermodynamic entropy definition, following a proposal by J. Barrett. Mathematical properties of this entropy are compared to physical consequences of the thought experiment. The validity of the second law of thermodynamics is investigated. In that cont…
Interplay between Standard Quantum Detailed Balance and Thermodynamically Consistent Entropy Production
The University of Tokyo（东京大学）；RIKEN Hakubi Research Team, Pioneering Research Institute (PRI), RIKEN（理化学研究所前沿研究机构（PRI）Hakubi研究团队）；Universal Biology Institute, The University of Tokyo（东京大学全球生物学研究院）；Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息研究中心）；Department of Physics, Graduate School of Science, The University of Tokyo（东京大学理学研究科物理学系）
Despite being a well-established operational approach to quantify entanglement, Rényi entropy calculations have been plagued by their computational complexity. We introduce here a theoretical framework based on an optimal thermodynamic integration scheme, where the Rényi entropy can be efficiently evaluated using regularizing paths. This approach avoids slowly convergent fluctuating contributions and leads to low-variance estimates. In this way, large system sizes and high levels of entanglement in model or first-principles Hamiltonians are within our reach. We demonstrate it in the one-dimens…
We take up the question why the initial entropy in the universe was small, in the context of evolution of the entropy of a classical system. We note that coarse-graining is a an important aspect of entropy evaluation which can reverse the direction of the increase in entropy, i.e., the direction of thermodynamic arrow of time. Then we investigate the role of decoherence in the selection of coarse-graining and explain how to compute entropy for a decohered classical system. Finally, we argue that the requirement of low initial entropy imposes constraints on the decoherence process.
By using entropy and entropy production, we calculate the steady flux of some phenomena. The method we use is a competition method, $S_S/τ+σ={\it maximum}$, where $S_S$ is system entropy, $σ$ is entropy production and $τ$ is microscopic interaction time. System entropy is calculated from the equilibrium state by studying the flux fluctuations. The phenomena we study include ionic conduction, atomic diffusion, thermal conduction and viscosity of a dilute gas.
We are discussing a universal non-unitary map M subsequent to a generic unitary map U, whose von Neumann entropy gain coincides with the calculated thermodynamic entropy production. For many-body quantum reservoirs we prove that M can be the averaging over all translations of the spatial frame. Assuming the coincidence of microscopic and macroscopic entropy productions leads to a novel equation between entropy gain of frame averaging and relative entropy. Our map M turns out to coincide with the older one called twirl, used recently in the theory of quantum reference frames. Related results to…
