The classical Boltzmann Bridge describes entropy histories conditioned on both an initial low-entropy macrostate and a later macrostate. Unlike the usual past-only formulation of the thermodynamic arrow, this two-time conditioning can produce entropy profiles that rise above the final entropy and then decrease toward the imposed endpoint. In this work, we formulate closed quantum analogues of the Boltzmann Bridge using macro-subspace projectors, unitary time evolution, and Boltzmann entropy defined by the dimension of coarse-grained macroscopic sectors. We first study a minimal coherent chamber-qubit model, in which each particle has only a two-state chamber degree of freedom. Although this model is the most direct quantization of the classical two-box system, its bridge entropy profile is dominated by coherent oscillations and revivals rather than classical relaxation. We then introduce a hidden-microstate bridge, in which each chamber sector contains unresolved internal degrees of freedom while the full dynamics remain unitary. Numerical experiments show that increasing the internal Hilbert-space dimension suppresses sample-dependent revival behavior and produces bridge entropy profiles whose sign structure and coarse-grained shape increasingly agree with the classical Boltzmann Bridge. We further use a Random Forest classifier to explore the parameter regime separating revival-dominated quantum behavior from classical-like coarse-grained bridge behavior. These results suggest that classical two-time-conditioned entropy behavior is not recovered by quantizing the chamber variable alone, but can emerge statistically from closed quantum.
The classical Boltzmann Bridge describes entropy histories conditioned on both an initial low-entropy macrostate and a later macrostate. Unlike the usual past-only formulation of the thermodynamic arrow, this two-time conditioning can produce entropy profiles that rise above the final entropy and then decrease toward the imposed endpoint. In this work, we formulate closed quantum analogues of the Boltzmann Bridge using macro-subspace projectors, unitary time evolution, and Boltzmann entropy defined by the dimension of coarse-grained macroscopic sectors. We first study a minimal coherent chamber-qubit model, in which each particle has only a two-state chamber degree of freedom. Although this model is the most direct quantization of the classical two-box system, its bridge entropy profile is dominated by coherent oscillations and revivals rather than classical relaxation. We then introduce a hidden-microstate bridge, in which each chamber sector contains unresolved internal degrees of freedom while the full dynamics remain unitary. Numerical experiments show that increasing the internal Hilbert-space dimension suppresses sample-dependent revival behavior and produces bridge entropy profiles whose sign structure and coarse-grained shape increasingly agree with the classical Boltzmann Bridge. We further use a Random Forest classifier to explore the parameter regime separating revival-dominated quantum behavior from classical-like coarse-grained bridge behavior. These results suggest that classical two-time-conditioned entropy behavior is not recovered by quantizing the chamber variable alone, but can emerge statistically from closed quantum.
The definition of entropy in both classical and quantum contexts relies on a coarse-graining of the underlying microscopic description. While this dependence is widely acknowledged, the physical constraints that determine which coarse-grainings are admissible are rarely made explicit. This issue becomes particularly acute when the universe is treated as a closed system, where no external observer or privileged level of description can be assumed. Any characterization of entropy — including claims about the low-entropy initial conditions of the universe — must then be grounded in descriptions that can be physically instantiated from within the system itself. In this work, we introduce a rigorous operational framework that establishes explicit physical selection principles over the space of admissible descriptions. We organize these criteria into necessary conditions (local accessibility and physical implementation via internal interactions) and practical sufficiency conditions (finite resource requirements anchored in Landauer’s principle, dynamical stability, and effective consistency). To demonstrate the validity of our framework, we introduce a concrete one-dimensional spin-chain toy model, illustrating how mathematically valid global partitions fail to be physically realizable due to dynamic propagation limits and informational constraints. We analyze the far-reaching implications of this restriction across three domains. In classical statistical mechanics, it limits the arbitrariness of macrostate definitions. In quantum theory, it identifies environment-induced decoherence (einselection) not merely as a formal tool, but as the objective dynamical mechanism that forces the emergence of admissible internal partitions. In cosmology, it resolves foundational circularities in the quasi-classical histories framework and reframes the thermodynamic arrow of time as a "doubly relational" property—dependent both on cosmic boundary conditions and on the material limits of internal subsystems. Our framework clarifies the operational status of entropy as an emergent property of physically instantiated descriptions within a universe that observes itself.
We consider a closed macroscopic quantum system in a pure state ψ evolving unitarily with Hamiltonian H in a finite-dimensional Hilbert space H. Following von Neumann, we assume a decomposition of H into mutually orthogonal subspaces Hν, where each ν represents a macro state, and Hν contains the pure states compatible with ν. We define the quantum Boltzmann entropy for a macro state ν as SqB(ν)=kBlogdimHν, where kB is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that SqB predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of H and Hν.
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.
This paper presents the definitive thermodynamic formulation of the recently established unified quantum phase space framework. By operating within a strictly positive joint distribution that unifies coordinate and momentum representations without operator expansions, we introduce a novel formulation of quantum entropy based on the classical Shannon-Boltzmann approach. We demonstrate that for any pure or mixed state, the total phase-space entropy decouples into the linear sum of structural spatial and momentum entropies. Utilizing the recently derived equation of motion for hybrid flow, we expose a purely internal, structural mechanism for quantum irreversibility and the arrow of time, arising directly from Fourier duality and phase-space lattice dynamics rather than external environmental coupling. The dynamic behavior of entropy production is analyzed. We provide compact, rigorous quantitative validations by demonstrating how free wave-packet dispersion represents natural decoherence and how the maximization of hybrid entropy naturally yields the classical-quantum Boltzmann-Gibbs distribution. These results establish a stable bridge between quantum mechanics, information theory, and macroscopic thermodynamics.
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,
Our everyday descriptions of the universe are highly coarse grained, following only a tiny fraction of the variables necessary for a perfectly fine-grained description. Coarse graining in classical physics is made natural by our limited powers of observation and computation. But in the modern quantum mechanics of closed systems, some measure of coarse graining is inescapable because there are no nontrivial, probabilistic, fine-grained descriptions. This essay explores the consequences of that fact. Quantum theory allows for various coarse-grained descriptions, some of which are mutually incompatible. For most purposes, however, we are interested in the small subset of “quasiclassical descriptions” defined by ranges of values of averages over small volumes of densities of conserved quantities such as energy and momentum and approximately conserved quantities such as baryon number. The near-conservation of these quasiclassical quantities results in approximate decoherence, predictability, and local equilibrium, leading to closed sets of equations of motion. In any description, information is sacrificed through the coarse graining that yields decoherence and gives rise to probabilities for histories. In quasiclassical descriptions, further information is sacrificed in exhibiting the emergent regularities summarized by classical equations of motion. An appropriate entropy measures the loss of information. For a “quasiclassical realm” this is connected with the usual thermodynamic entropy as obtained from statistical mechanics. It was low for the initial state of our universe and has been increasing since.