One of the major unsolved problems in physics is the contradiction betwee n the reversibility of the microscopic world, as described by quantum mech anics, and the irreversibility of the macroscopic world, as described by c lassical thermodynamics and statistical mechanics.Specifically, time evol ution governed by the Schrödinger equation is unitary and possesses time-r eversal symmetry, and the von Neumann entropy of a microscopic system is c onserved.However, the entropy increase observed in macroscopic, isolated systems according to the second law of thermodynamics cannot be explained by unitary time evolution and von Neumann entropy alone.This doctoral dissertation presents a unified framework that uses observe rs to explain the contradiction.Based on von Neumann's concept of macrosc opic entropy, the framework redefines these concepts through the lens of q uantum information theory and specifically, Petz's theory of quantum suffi ciency and resource theory.This research proposes that macroscopic irreversible phenomena do not vio late physical laws.Instead, these phenomena result from the loss of stati stical information caused by limitations in an observer's measurement capa bilities or by coarsening.This doctoral thesis specifically focuses on ob servational entropy, which is a generalization of von Neumann's macroscopi c entropy.Chapter 4 of this study reinterprets von Neumann's H-theorem within a con temporary framework.The chapter provides a mathematical proof of the irre versible increase of observational entropy in isolated quantum systems.It was demonstrated that, when the system's initial state is macroscopic, or lacking microscopic details with respect to a specific measurement, obser Macroscopic States, Operations, and Entropy: An Obser ver-Centric Approach (巨視的な状態、操作、エントロピー ：観測者中心のアプローチ) 長澤 輝明
One of the major unsolved problems in physics is the contradiction betwee n the reversibility of the microscopic world, as described by quantum mech anics, and the irreversibility of the macroscopic world, as described by c lassical thermodynamics and statistical mechanics.Specifically, time evol ution governed by the Schrödinger equation is unitary and possesses time-r eversal symmetry, and the von Neumann entropy of a microscopic system is c onserved.However, the entropy increase observed in macroscopic, isolated systems according to the second law of thermodynamics cannot be explained by unitary time evolution and von Neumann entropy alone.This doctoral dissertation presents a unified framework that uses observe rs to explain the contradiction.Based on von Neumann's concept of macrosc opic entropy, the framework redefines these concepts through the lens of q uantum information theory and specifically, Petz's theory of quantum suffi ciency and resource theory.This research proposes that macroscopic irreversible phenomena do not vio late physical laws.Instead, these phenomena result from the loss of stati stical information caused by limitations in an observer's measurement capa bilities or by coarsening.This doctoral thesis specifically focuses on ob servational entropy, which is a generalization of von Neumann's macroscopi c entropy.Chapter 4 of this study reinterprets von Neumann's H-theorem within a con temporary framework.The chapter provides a mathematical proof of the irre versible increase of observational entropy in isolated quantum systems.It was demonstrated that, when the system's initial state is macroscopic, or lacking microscopic details with respect to a specific measurement, obser Macroscopic States, Operations, and Entropy: An Obser ver-Centric Approach (巨視的な状態、操作、エントロピー ：観測者中心のアプローチ) 長澤 輝明
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,
Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of mixing quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit in quantum statistical mechanics and quantum gravity. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.
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.
For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
We construct a macroscopic thermodynamical theory that satisfies the second law of thermodynamics, based on quantum mechanics. More concretely, our macroscopic theory satisfies the following two features: (1) A macroscopic equilibrium state is expressed as a set of macroscopic variables a=(a1,⋯,aL), which represents coarse-grained values of a set of physical quantities, and (2) An adiabatic transformation from an equilibrium state a to another state a′ is possible if and only if sa≤sa′ holds. Here, sa is a real-valued function corresponding to the thermodynamical entropy, which depends only on macroscopic variables a. We show that such a macroscopic thermodynamical theory can be consistently constructed if the von-Neumann entropy of a generalized microcanonical state satisfies a generalized central limit theorem. We also show that even when the von-Neumann entropies do not satisfy the condition, a macroscopic thermodynamic-like theory with a similar structure exists, which could be at the foundations of nonequilibrium thermodynamics. Finally, we make a replica of (U,V,N) expression of thermodynamics satisfying (1) and (2), using a toy model of free particles in a box.
The evident contrast between the time symmetry of fundamental microscopic laws and the time asymmetry of macroscopic processes is a challenging physical problem. The observation of unitary evolution of a general physical system by an idealized observer whose recording and processing information is time neutral is studied in a full information context. In the macroscopic limit, a contradiction between observations and predictions done at different times appears. It implies that the observer cannot read records performed at times when the entropy of the system is larger.