Macroscopic fields such as electromagnetic, magnetohydrodynamic, acoustic or gravitational waves are usually described by classical wave equations with possible additional damping terms and coherent sources. The aim of this paper is to develop a complete macroscopic formalism including random/thermal sources, dissipation and random scattering of waves by environment. The proposed reduced state of the field combines averaged field with the two-point correlation function called single-particle density matrix. The evolution equation for the reduced state of the field is obtained by reduction of the generalized quasi-free dynamical semigroups describing irreversible evolution of bosonic quantum field and the definition of entropy for the reduced state of the field follows from the von Neumann entropy of quantum field states. The presented formalism can be applied, for example, to superradiance phenomena and allows unifying the Mueller and Jones calculi in polarization optics.
Macroscopic fields such as electromagnetic, magnetohydrodynamic, acoustic or gravitational waves are usually described by classical wave equations with possible additional damping terms and coherent sources. The aim of this paper is to develop a complete macroscopic formalism including random/thermal sources, dissipation and random scattering of waves by environment. The proposed reduced state of the field combines averaged field with the two-point correlation function called single-particle density matrix. The evolution equation for the reduced state of the field is obtained by reduction of the generalized quasi-free dynamical semigroups describing irreversible evolution of bosonic quantum field and the definition of entropy for the reduced state of the field follows from the von Neumann entropy of quantum field states. The presented formalism can be applied, for example, to superradiance phenomena and allows unifying the Mueller and Jones calculi in polarization optics.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
While the dynamic state of a single particle can be specified quantum mechanically in terms of its state function, any rigorous description of the state of a many-particle system would require the complete knowledge of the dynamic state functions of all the particles. That is not always possible. On the other hand, for a large number of particles in, or near, thermal equilibrium in a uniform sample, the principles of statistical mechanics may be invoked to describe the averaged expectation values of the physically observable properties of such a many-particle system. The basic concepts of the density-matrix formalism and the quantum mechanic analog of the classical Boltzmann equation commonly used for optical and magnetic resonance problems of many-particle quantum systems are introduced in this chapter. Applications of this approach to such specific problems as the resonant interaction of electromagnetic radiation with optical media of two-level atoms, nonlinear optics, and the laser rate equations and transient dynamics are discussed in this chapter. Definitions of the density operator and the density matrix Up to this point, in studying the dynamics of quantum mechanic systems, we have assumed that the state of the system can be specified in terms of a precisely known state function |Ψ〉. On the other hand, for a macroscopic medium containing many particles, it is not always possible to know the exact dynamic states of all the particles in the medium, even for physically identical particles.
We discuss compatibility between various quantum aspects of bosonic fields, relevant for quantum optics and quantum thermodynamics, and the mesoscopic formalism of reduced state of the field (RSF). In particular, we derive exact conditions under which Gaussian and Bogoliubov-type evolutions can be cast into the RSF framework. In that regard, special emphasis is put on Gaussian thermal operations. To strengthen the link between the RSF formalism and the notion of classicality for bosonic quantum fields, we prove that RSF contains no information about entanglement in two-mode Gaussian states. For the same purpose, we show that the entropic characterisation of RSF by means of the von Neumann entropy is qualitatively the same as its description based on the Wehrl entropy. Our findings help bridge the conceptual gap between quantum and classical mechanics.
The Schrodinger equation with Gaussian dynamic fluctuating potentials is considered. This model gives a convenient example of an open quantum system coupled to a given external environment. The evolution equations for averaged correlators are derived and it is shown that a system coupled to a thermoequilibrium environment tends asymptotically with time to a thermodynamic density matrix. At high temperatures the quantum evolution equation for the density matrix can be approximated by the quasiclassical Fokker-Planck equation. The generalisation for multiparticle systems is also given. The results are illustrated by the particular examples of coupling with phonons and of two-level systems. Finally, it is shown how the problem can be reformulated in terms of continuous stochastic evolution in the space of psi functions and a corresponding functional formalism based on the continuous Fokker-Planck equation in the space of psi functions.
The goal of this thesis is to define the concept of coarse-graining in quantum systems through measurements and to develop a unified mathematical framework for macroscopic states, operations, correlations, and entropy.Within this framework, irreversible phenomena arise from the loss of statistical information that is accessible to macroscopic observers.To achieve this goal, we rigorously define macroscopicity.This thesis investigates observational entropy, a concept that unifies Boltzmann entropy, Gibbs entropy, von Neumann's macroscopic entropy, and diagonal entropy.Our approach to explaining how irreversibility arises in isolated quantum systems is based on this unified perspective.When the initial state is "macroscopic," we use Petz's theory of statistical sufficiency to show that observational entropy strictly increases with unitary time evolution.Furthermore, we show that, for any initial state, the probability of the system's observational entropy increasing and rapidly reaching its maximum value under random unitary time evolution is extremely high.Specifically, we demonstrate that when measurements on a quantum system are sufficiently coarse-grained relative to its size, its initial state (whether pure or mixed) becomes indistinguishable from the maximally mixed state through random time evolution.We show that this holds not only for random time evolutions sampled according to the Haar measure, but also for approximate 2-designs, which are considered more physically realistic models.Next, we extend this framework by introducing positive operator-valued measures (POVMs) and quantum prior states.This allows us to construct a unified inferential framework for macroscopic states.A macroscopic state can be characterized using equivalent algebraic and constructive formulations based on quantum statistical sufficiency.This approach is based on the concept of an observational deficit, which quantifies the degree to which a state is "irretrodictable" relative to prior and measurement.This structure is formulated as an "inferential reference frame" that encapsulates the observer's informational perspective.Furthermore, we treat macroscopic states as free states and construct the resource theory of microscopicity.This theory unifies existing theories of coherence, athermality and asymmetry.Finally, we extend this framework to the study of quantum correlations under observational constraints.This enables us to show that quantum correlations, such as entanglement and quantum discord, are context-dependent resources that depend on the observer's reference frame.I would like to express my deepest gratitude to my supervisor,
Local properties of a quantum system are defined as the expectation values of its observables in a microstate of some complete set of commuting observables. An equation for the time evolution of local properties is obtained for any system whose statistical operator (density matrix) evolves by unitary transformation in accordance with the von Neumann equation. The formalism is applied to the example of a system of one particle. In this case the local properties are fields, the time-evolution equation is an equation of continuity with source terms. For constants of motion the source terms vanish, giving equations of continuity for the fields. For each scalar field a flux vector for its transport current is defined. For momentum, a stress tensor is obtained. The effects on local properties of realization of a latent ensemble of the statistical operator (an entropy-increasing mechanism recently proposed to explain approach to equilibrium) are also considered; a non-negative local entropy production is identified, as well as a discontinuous redistribution of local properties among microstates.