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Journal of Mathematical Physics

On the relation between classical and quantum-thermodynamic entropy

Gian Paolo Beretta

Published May 1, 1984• 19 citations
Publisher

Abstract

We discuss the unresolved problem of proving rigorously that in the classical limit ℏ→0, the quantum-thermodynamic entropy functional tends to the classical entropy functional. We state rather restrictive conditions that define the general problem of finding a complete classical phase-space representation of quantum kinematics. Whether the problem admits of solutions remains an unresolved question. We discuss a physically interesting attempt to relate the structure in the classical limit ℏ→0 of the well-known Blokhintzev, Wigner, and Wehrl phase-space functions to the spectral expansion of the quantum state operator.

DOI
10.1063/1.526322

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We discuss the unresolved problem of proving rigorously that in the classical limit ℏ→0, the quantum-thermodynamic entropy functional tends to the classical entropy functional. We state rather restrictive conditions that define the general problem of finding a complete classical phase-space representation of quantum kinematics. Whether the problem admits of solutions remains an unresolved question. We discuss a physically interesting attempt to relate the structure in the classical limit ℏ→0 of the well-known Blokhintzev, Wigner, and Wehrl phase-space functions to the spectral expansion of the quantum state operator.
The concept of quantum phase space offers a view on quantum mechanics, which is different from the standard Hilbert space approach, but which more closely resembles the classical phase space. Due to the properties of quantum mechanics there are several equivalent quantum phase space descriptions, and one cannot always prefer one or another as they all have certain merits and drawbacks. For example, the Husimi-Kano Q function is a probability distribution and thus gives rise to entropic quantities, namely the Renyi-Wehrl entropies, of which several properties are known. The Wigner function, on the other hand, has an easier physical explanation, but may take negative values. In this article, we investigate entropic measures of localisation for a state in quantum phase space by using the Beckner-Brascamp-Lieb inequality to relate different phase-space functions.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its ℏ\hbarℏ-dependent ⋆\star⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.
We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $ln\ensuremath{\pi}+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics where it allows us to establish a Wigner entropy-power inequality. It also opens a way towards stronger entropic uncertainty relations. Finally, we define the Wigner-R\'enyi entropy of Wigner-positive states and conjecture an extended lower bound that is reached for Gaussian pure states.
The study of the quantum–classical correspondence has been focused on the quantum measurement problem. However, most of the discussion in the preceding chapters is motivated by a broader question: Why do we perceive our quantum Universe as classical? Therefore, emergence of the classical phase space and Newtonian dynamics from the quantum Hilbert space must be addressed. Chapter 6 starts by re-deriving decoherence rate for non-local superpositions using the Wigner representation of quantum states. We then discuss the circumstances that, in some situations, make classical points a useful idealization of the quantum states of many-body systems. This classical structure of phase space emerges along with the (at least approximately reversible) Newtonian equations of motion. Approximate reversibility is a non-trivial desideratum given that the quantum evolution of the corresponding open system is typically irreversible. We show when such approximately reversible evolution is possible. We also discuss quantum counterparts of classically chaotic systems and show that, as a consequence of decoherence, their evolution tends to be fundamentally irreversible: They produce entropy at the rate determined by the Lyapunov exponents that characterize classical chaos. Thus, quantum decoherence provides a rigorous rationale for the approximations that led to Boltzmann’s H-theorem.
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.