🔬 condensed matter

Von Neumann's Two Quantum Entropies

This paper argues that while the well-known 1927 von Neumann entropy is merely a measure of entanglement, the overlooked 1929 entropy is the true thermodynamic entropy for quantum states.

Original authors: Biao Wu

Published 2026-09-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Biao Wu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Entropy is the measure of disorder in a system, the reason why a cup of coffee cools down and why a shattered glass never spontaneously reassembles. For over a century, physicists have sought to understand how this familiar, messy behavior emerges from the clean, precise laws that govern individual atoms. When the rules of the quantum world were discovered in the early twentieth century, a new question arose: how do we define this disorder for a single quantum particle? The answer seemed straightforward at first, but a deeper look reveals that the definition most scientists have used for nearly a hundred years might be measuring the wrong thing entirely.

In 1927, the legendary physicist John von Neumann proposed a way to calculate the entropy of a quantum system. His formula, now known as the von Neumann entropy, became a cornerstone of quantum physics. It works beautifully for describing how different parts of a quantum system become linked, a phenomenon called entanglement. However, the same formula has a strange flaw when applied to the heat and disorder of a real-world object. If a quantum system is in a "pure" state—a condition where it is perfectly defined and not mixed with anything else—this famous formula says its entropy is exactly zero. This creates a paradox. In the real world, even a perfectly prepared gas of atoms will eventually spread out and mix, a process that clearly increases disorder. Yet, according to the 1927 formula, a pure quantum system never changes its entropy; it stays at zero forever, regardless of how much it spreads or mixes.

Decades later, in 1929, von Neumann realized this problem and proposed a different formula to fix it. He introduced a second type of entropy, which we can call the Wigner–von Neumann entropy. Unlike the first formula, this one does not stay at zero for pure states. Instead, it increases as the system evolves, capturing the very real, irreversible spreading of particles that we see in nature. Despite being the better tool for describing thermodynamic heat and disorder, this second formula was largely forgotten. For nearly a century, the scientific community continued to use the 1927 version for almost everything, treating it as the universal measure of quantum disorder, while the 1929 version sat in the archives, ignored and unappreciated.

A recent paper by physicist Biao Wu seeks to correct this historical oversight. The author argues that we have been using the wrong tool for the job. The study reviews the history and mathematical properties of both formulas to show that they were designed for two completely different purposes. The 1927 entropy, the author concludes, is not a measure of thermodynamic heat at all. Instead, it is a precise measure of entanglement, the invisible link between quantum particles. It tells us how much information is shared between two parts of a system, but it fails to tell us anything about the heat or disorder of the system as a whole.

The paper demonstrates that the 1929 entropy is the true thermodynamic entropy for quantum systems. To prove this, the author examines a simple scenario: a gas of particles confined to one side of a container that is then allowed to spread out and fill the entire space. This is a classic irreversible process where disorder increases. The 1927 formula sees this process and reports no change, because the quantum state of the gas remains pure and the formula is blind to the spreading. In contrast, the 1929 formula correctly registers an increase in entropy, mirroring the behavior of a real gas. It accounts for the fact that while we might know the exact quantum state of the system, we, as macroscopic observers, can only measure a limited set of properties, such as position and momentum. The 1929 formula is built on this limitation, measuring the disorder we actually experience.

One of the most compelling arguments in the paper is that the 1929 entropy behaves like a true thermodynamic quantity should: it is extensive. This means that if you combine two separate systems, the total entropy is simply the sum of their individual entropies. The 1927 formula does not always do this, which is a major reason why it cannot represent the heat and disorder of a large collection of atoms. The author also addresses why the 1929 formula was forgotten in the first place. It required a specific way of dividing the quantum world into small, manageable chunks, a method that seemed difficult to calculate and implement. However, modern computational techniques have solved these technical hurdles, making it possible to calculate this entropy efficiently.

The paper suggests that the reason the 1927 formula worked so well in the past is a mathematical coincidence. In many common experiments, a small system interacts with a much larger environment, like a heat bath. In these specific situations, the two formulas happen to give the same numerical result. This led scientists to believe they were interchangeable. The new analysis shows that this agreement is an accident of the specific conditions, not a sign that the formulas are the same. When the conditions change, as they do in the spreading gas example, the formulas diverge, and only the 1929 version tells the truth about the physical world.

Ultimately, this work is a call to reorganize how we think about quantum disorder. It proposes a clear division of labor: use the 1927 formula when you want to understand the mysterious connections between quantum particles, and use the 1929 formula when you want to understand heat, temperature, and the irreversible flow of time. By reviving the forgotten 1929 entropy, the paper restores the correct link between the microscopic quantum world and the macroscopic thermodynamic world we live in, ensuring that our mathematical tools match the physical reality they are meant to describe.

Technical Summary: Von Neumann's Two Quantum Entropies

Problem Statement
The paper addresses a historical and conceptual ambiguity regarding the definition of thermodynamic entropy in quantum systems. John von Neumann proposed two distinct entropy measures in 1927 and 1929, both intended to define thermodynamic entropy for quantum states. The 1927 entropy (now known as the von Neumann or vN entropy, $S_v = -\text{Tr}(\hat{\rho} \ln \hat{\rho})$ ) is the standard measure in quantum information theory. However, the 1929 entropy (referred to here as the Wigner–von Neumann or WvN entropy, $S_w = -\sum p_\alpha \ln p_\alpha$ ), which relies on coarse-grained probability distributions over phase space, has been largely ignored. The central problem is determining which of these two quantities correctly represents the thermodynamic entropy observed in macroscopic systems, particularly given that $S_v$ vanishes for all pure states and remains invariant under unitary evolution, seemingly failing to capture the irreversible processes characteristic of thermodynamics.

Methodology and Historical Review
The author conducts a comprehensive review of the derivations and properties of both entropies:

Re-evaluation of the 1927 Derivation: The paper revisits von Neumann's 1927 thought experiment involving an ideal gas of fictitious molecules and semi-permeable walls. The author argues that this derivation equates quantum thermodynamic entropy with the mixing entropy of a classical gas. They highlight a critical physical limitation: constructing the semi-permeable walls required to measure the specific operator $\hat{R}$ distinguishing orthogonal states generally requires an exponentially large amount of physical resources, rendering the measurement physically unfeasible for general quantum systems.
Analysis of the 1929 Derivation: The author examines von Neumann's 1929 work, where he introduced the WvN entropy to prove a quantum H-theorem. This entropy is defined by partitioning phase space into Planck cells and assigning localized wave functions (basis functions) to them. The probability distribution is derived from the projection of the system's state onto these cells. The author notes that while von Neumann attributed this idea to Wigner, the technical issues regarding the choice of basis and the "cumbersome" nature of the calculation have since been resolved through efficient numerical methods (e.g., using Wannier functions) and the realization that coarse-graining is not strictly necessary for the H-theorem.
Comparison of Properties: The paper contrasts the mathematical behaviors of $S_v$ and $S_w$ . Specifically, $S_v$ is zero for pure states and invariant under unitary evolution, whereas $S_w$ is nonzero for almost all pure states and evolves under unitary dynamics.
Physical Basis Analysis: The author introduces the concept of a "physical basis," defined as a set of orthonormal functions for which corresponding measurements can be performed without exponentially large physical resources. They argue that the basis functions used in the WvN entropy (localized in position/momentum) correspond to macroscopic measurements (like those in cloud chambers) that are physically realizable, unlike the generic basis required for $S_v$ .

Key Contributions and Results

Reclassification of Entropy Roles: The primary contribution is the argument that the 1927 vN entropy should not be viewed as thermodynamic entropy but rather as a measure of quantum entanglement. This aligns with Everett's 1957 insight that the vN entropy quantifies the correlation (entanglement) between subsystems.
Identification of Thermodynamic Entropy: The paper asserts that the 1929 WvN entropy is the true thermodynamic entropy for quantum systems. Unlike $S_v$ , $S_w$ increases during irreversible processes such as gas diffusion, even for isolated systems evolving unitarily. It captures the spreading of the wave function in phase space (both spatial and momentum directions), thereby reflecting the macroscopic irreversibility observed in experiments.
Extensivity: The author demonstrates that the WvN entropy is extensive ( $S_{A+B}^w \approx S_A^w + S_B^w$ ) for large Hilbert spaces, a fundamental property of thermodynamic entropy that the vN entropy generally lacks.
Explanation of Historical Consistency: The paper explains why the use of vN entropy as thermodynamic entropy has not led to experimental contradictions. In the common scenario of a small system interacting with a large heat bath, the vN entropy of the small system ( $S_v^S$ ) and the WvN entropy ( $S_w^S$ ) are mathematically coincident ( $S_v^S \approx S_w^S \approx \ln D_S$ ), allowing both to yield the Boltzmann–Gibbs distribution via the maximum entropy principle.

Significance and Claims
The paper claims that the historical neglect of the 1929 entropy was due to technical difficulties in its computation and a lack of clarity regarding the choice of basis functions. By resolving these technical issues and re-evaluating the physical feasibility of the measurements required for each entropy, the author argues for a paradigm shift:

The vN entropy is the correct tool for quantifying entanglement in quantum information.
The WvN entropy is the correct tool for defining thermodynamic entropy in quantum statistical mechanics.

The author concludes that recognizing this distinction clarifies the microscopic origin of thermodynamic irreversibility in quantum systems and corrects a century-long conflation of entanglement measures with thermodynamic quantities. They note that the concept of defining entropy via a specific set of basis functions has been recently rediscovered, lending further support to their argument.

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