Abstract Using quantum information geometry, I derive quantum generalizations of the Onsager rate equations, which model the dynamics of an open system near a steady state. The generalized equations hold for a flexible definition of the forces as well as a large class of statistical divergence measures and quantum-Fisher-information metrics beyond the conventional definition of entropy production. I also derive quantum Onsager–Casimir relations for the transport tensors by proposing a general concept of time reversal and detailed balance for open quantum systems. The results establish a remarkable connection between statistical mechanics and parameter estimation theory.
Abstract Using quantum information geometry, I derive quantum generalizations of the Onsager rate equations, which model the dynamics of an open system near a steady state. The generalized equations hold for a flexible definition of the forces as well as a large class of statistical divergence measures and quantum-Fisher-information metrics beyond the conventional definition of entropy production. I also derive quantum Onsager–Casimir relations for the transport tensors by proposing a general concept of time reversal and detailed balance for open quantum systems. The results establish a remarkable connection between statistical mechanics and parameter estimation theory.
A quantum statistical expression for the entropy of a nonequilibrium system is defined so as to be consistent with Gibbs’ relation, and is shown to corresponds to dynamical variable by introducing analogous to the Heisenberg picture in quantum mechanics. The general relation between system-reservoir interactions and an entropy change operator in an open quantum system, relying just on the framework of statistical mechanics and the definition of von Neumann entropy. By using this formula, we can obtain the correct entropy production in the linear response framework. The present derivation of entropy production is directly based on the first principle of microscopic time-evolution, while the previous standard argument is due to the thermodynamic energy balance.
The analogue of the classical Onsager theory of entropy production is systematically derived for weakly irreversible processes in open quantum systems with finite-dimensional Hilbert space. The dynamics is assumed to be given by a quantum dynamical semigroup with infinitesimal generator of Gorini-Kossakowski-Sudarshan type. The basic Spohn formula for entropy production is used to obtain an expansion in terms of powers of the deviation of the initial state relative to the final stationary state of irreversible dynamics. To this end, an appropriate Lie series is constructed from a particular symmetrization procedure applied to the ordinary Campbell-Hausdorff expansion. In this way, only Hermitian contributions by higher-order commutators are generated, which allow an identification with so-called generalized Onsager coefficients. The explicit derivations concentrate on second-, third- and fourth-order coefficients, whereas complete detailed expressions are worked out for second and third order. In a suitable coherence-vector representation of density matrices the results can be given in terms of the dynamical parameters fixing the infinitesimal semigroup generator and in terms of symmetric and antisymmetric structure constants of the Lie algebra of SU ( N ). As an illustration, an application to generalized Bloch equations for two-level systems is studied, where the Onsager-like expansion can be compared with exact results for entropy production. We find that convergence is good even for rather large deviations between initial and final state if the calculation includes second- and third-order coefficients only. The formalism presented in this paper generalizes restrictions on admitted final states adopted in much simpler earlier treatments to the most general case of arbitrary unique final states of irreversible processes.
Abstract Processes at the nanoscale happen far away from the thermodynamic limit, far from equilibrium and are dominated by fluctuations and, perhaps, even quantum effects. This book establishes a consistent thermodynamic framework for such processes by combining tools from non-equilibrium statistical mechanics and the theory of open quantum systems. The book is accessible for graduate students and of interest to all researchers striving for a deeper understanding of the laws of thermodynamics beyond their traditional realm of applicability. It puts most emphasis on the microscopic derivation and understanding of key principles and concepts as well as their interrelation. The topics covered in this book include (quantum) stochastic processes, (quantum) master equations, local detailed balance, classical stochastic thermodynamics, (quantum) fluctuation theorems, strong coupling and non non-Markovian effects, thermodynamic uncertainty relations, operational approaches, Maxwell's demon and time-reversal symmetry, among other topics. Furthermore, the book treats a few applications in detail to illustrate the general theory and its potential for practical applications. These are single-molecule pulling experiments, quantum transport and thermoelectric effects in quantum dots, the micromaser and related set-ups in quantum optics.
This paper introduces a generalized definition of quantum information entropy, aiming to provide a more comprehensive framework for quantifying information in complex quantum systems. The core of this work lies in extending the conventional concept of entropy, traditionally applied to classical information, to the realm of quantum mechanics. We explore various types of quantum entropy, including von Neumann entropy, relative entropy, and a newly proposed generalized entropy, denoted as *Eg*. The definition of *Eg* is based on a novel approach that considers the entanglement structure and correlations within quantum states. We establish relationships between these different entropy measures, demonstrating their interconnectedness within the broader context of quantum information theory. The paper further outlines a mathematical formalism for calculating *Eg*, providing a tool for analyzing and characterizing quantum information. Our generalized entropy provides a more flexible and potentially more accurate measure of quantum information, particularly in scenarios involving complex quantum correlations and multi-partite quantum systems. ---
We analyze manifestations of so-called information entropy production and the role of information dynamics in the Schrödinger picture quantum evolution. PACS numbers: 03.65.-w, 02.50.-r In a number of recent papers, [1]-[5] c.f. also [6]-[13], various links were disclosed between the concept of Fisher information in statistical inference theory (plus related indeterminacy-type inequalities) and localization properties of Schrödinger wave functions. Except for a brief notice in Ref. [13], no attention has been paid to the full-fledged Schrödinger quantum evolution and to the fact that a central role in the formalism is in fact played not by the Fisher information, but by the (formally related) concept of (configurational) information entropy production, which appears to be inseparably tied with the dynamics of quantum probability densities and the persistence of steady (stationary) states. Information theory concepts are normally introduced in the context of equilibrium and nonequlibrium statistical mechanics as natural tools for quantifying phase-space disorder (dynamical origin of increasing entropy, irreversibility and time arrow issues). However, their range of applicability
A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the Onsager–Casimir reciprocity relations to the nonlinear response coefficients.