The second law of thermodynamics is one of the fundamental principles of nature, often summed up by the notion that entropy—a measure of disorder—always increases in a closed system over time. At its core, the law suggests that systems evolve towards greater disorder. Ice melts into water, porcelain breaks into shards, and ordered structures inevitably lose their organization. However, when we turn our attention to quantum physics, we encounter an apparent contradiction. In quantum systems, entropy, as defined by conventional mathematical models, seems to remain unchanged.
This paradox has puzzled scientists for years, as it suggests that quantum mechanics might violate a fundamental principle of thermodynamics. However, a new study from a research team at TU Wien sheds light on this issue, showing that the apparent conflict can be resolved by redefining the concept of entropy. When entropy is understood in a way that is compatible with quantum mechanics, it becomes clear that, just like in classical thermodynamics, entropy increases in quantum systems as well, eventually reaching a state of disorder.
The research, published in PRX Quantum, demonstrates that, when approached correctly, the second law of thermodynamics holds true in the quantum realm.
Understanding Entropy: Beyond Disorder
First, it’s crucial to understand what entropy means in both classical and quantum contexts. The term “entropy” is often associated with disorder, but this is a simplification. The second law of thermodynamics tells us that in isolated systems, entropy increases, leading to a more probable state of greater disorder. But how exactly do we measure and define entropy?
According to Prof. Marcus Huber from TU Wien’s Institute for Atomic and Subatomic Physics, entropy is fundamentally a measure of how specific or random the states of a system are. If a system is in a highly ordered, specific state, its entropy is low. If the system is in a state with many possible configurations that look more or less the same, then the entropy is high.
For example, consider a box full of balls that are sorted by color. If you shake the box, the balls will mix together, resulting in a higher entropy state. The key here is that there are only a few ordered configurations but many disordered configurations. This leads to a natural increase in entropy when the system moves towards a more disordered state.
This process of increasing entropy, according to classical thermodynamics, defines the direction of time. As Max Lock (TU Wien) explains, the past is characterized by lower entropy, while the future is associated with higher entropy. In other words, time moves forward as entropy increases.
The Quantum Paradox: von Neumann’s Insight
While the second law of thermodynamics holds in the classical world, things become less clear in the quantum realm. Quantum systems are governed by the laws of quantum mechanics, which include the von Neumann entropy—a mathematical measure that quantifies the entropy of a quantum system.
According to John von Neumann, one of the pioneers of quantum mechanics, the von Neumann entropy remains unchanged over time. This creates a paradox. If the entropy of a quantum system does not change, then it would be impossible to differentiate between the past and the future. Time, in this view, could move backward or forward with no difference, which contradicts our everyday experience of a world where disorder naturally increases.
In essence, if we apply von Neumann’s entropy to a closed quantum system, we find that the system’s entropy doesn’t change over time, making the concept of increasing entropy (and thus the arrow of time) seem irrelevant in quantum systems.
A New Approach: Limited Information in Quantum Systems
However, the TU Wien research team, led by Tom Rivlin and others, realized that the von Neumann entropy approach overlooked something important: in quantum mechanics, we never have full information about a system. Unlike classical systems where we can know the complete state, in quantum physics, we can only measure certain properties of the system—such as the position or momentum of a particle. This is a key distinction because, in quantum theory, measurement results are governed by probabilities, and we can never be certain of the outcome before performing the measurement.
For example, when measuring the position of a particle, we know the probabilities of where it might be found, but the actual outcome remains uncertain until the measurement is made. This uncertainty means that we need to define entropy in a way that incorporates this element of surprise.
Instead of using von Neumann entropy, which applies to the entire system, the researchers propose a new concept of entropy based on Shannon entropy, which takes into account only the probabilities of measurement outcomes. This type of entropy is tied to the information content of the system. As Florian Meier (TU Wien) explains, Shannon entropy measures how much new information we gain when we make a measurement. If we are sure of the result (i.e., if there is only one possible measurement outcome), then the Shannon entropy is zero. But if there are many possible outcomes, each with a high probability, the entropy is large.
The Resolution: Entropy Increases in Quantum Systems
Using this new understanding of entropy, the TU Wien team was able to show that Shannon entropy behaves just like classical entropy in quantum systems. If you start with a state of low entropy—say, an ordered quantum system—the entropy increases over time, leading to greater uncertainty and surprise in measurement outcomes. Eventually, the system reaches a state of maximum entropy, just as classical systems do.
This means that the second law of thermodynamics, which states that entropy increases over time, also applies in closed quantum systems. However, to observe this increase in entropy, we must use the right definition of entropy—Shannon entropy—instead of von Neumann entropy.
This conclusion was confirmed by both mathematical proof and computer simulations, which showed that entropy increases in quantum systems composed of multiple interacting particles. These findings are significant because they resolve the apparent contradiction between quantum physics and thermodynamics, showing that the laws governing entropy hold true even in the quantum world.
Implications for Quantum Technologies
This breakthrough is not just a theoretical curiosity. In the world of modern quantum technologies, such as quantum computing and quantum information processing, we often work with systems that contain many interacting quantum particles. For these systems, reconciling quantum mechanics with thermodynamic principles is essential for understanding their behavior and for developing efficient quantum technologies.
As Marcus Huber explains, while small quantum systems, like a single atom, may not exhibit these entropy behaviors in a noticeable way, large quantum systems—such as those used in quantum computers—require a deeper understanding of how quantum systems evolve and interact over time. The work done by the TU Wien team lays the groundwork for future advancements in quantum thermodynamics, which is crucial for the development of reliable and scalable quantum technologies.
Conclusion: Quantum Systems and the Arrow of Time
The second law of thermodynamics has been a cornerstone of our understanding of the physical world for more than a century. It tells us that entropy always increases, moving systems toward greater disorder. While quantum mechanics initially seemed to defy this principle, recent research has shown that when we define entropy in a way that accounts for the uncertainty and probabilistic nature of quantum measurements, entropy still increases in quantum systems.
This discovery resolves the apparent conflict between quantum mechanics and thermodynamics, showing that the arrow of time—the unidirectional flow from order to disorder—holds true even in the quantum realm. For scientists working on the cutting edge of quantum technology, this new understanding provides essential insights into how quantum systems behave and offers a pathway for the development of future quantum technologies. The study not only strengthens our understanding of the quantum world but also bridges the gap between quantum theory and classical thermodynamics, offering a unified view of the universe at both the microscopic and macroscopic scales.
Reference: Florian Meier et al, Emergence of a Second Law of Thermodynamics in Isolated Quantum Systems, PRX Quantum (2025). DOI: 10.1103/PRXQuantum.6.010309