Shattering the Quantum Entropy Myth – Why Disorder Always Wins

Entropy always increases, but in quantum systems, traditional entropy measures seem constant. TU Wien researchers resolved this paradox by considering Shannon entropy, which accounts for the uncertainty in measurements.

Their work proves that even in isolated quantum systems, disorder naturally grows, aligning quantum mechanics with thermodynamics.

The Paradox of Entropy in Quantum Physics

The second law of thermodynamics is one of the fundamental principles of nature. It states that in a closed system, entropy — the measure of disorder — must always increase over time. This explains why structured systems naturally break down: ice melts into water, and a shattered vase will never reassemble itself. However, quantum physics appears to challenge this rule. Mathematically, entropy in quantum systems seems to remain unchanged, raising a puzzling contradiction.

A research team at TU Wien investigated this apparent inconsistency and found that the answer depends on how entropy is defined. When entropy is measured in a way that aligns with quantum mechanics, the contradiction disappears. Just like in classical physics, entropy in quantum systems increases, leading initially ordered systems toward disorder.

Entropy and the Direction of Time

While entropy is often associated with disorder, the two are not identical. What seems disordered can be subjective, but entropy has a precise mathematical definition. It quantifies whether a system is in a highly specific state (low entropy) or one of many possible states that appear similarly random (high entropy).

“Entropy is a measure of whether a system is in a special, very particular state, in which case the system has low entropy, or whether it is in one of many states that look more or less the same, in which case it has high entropy,” explains Prof Marcus Huber from the Institute for Atomic and Subatomic Physics at TU Wien. If you start with a very specific state, for example, a box full of balls that are sorted exactly by color, then if you shake the box a little, a higher entropy mixed state will develop over time. This is simply due to the fact that only a few ordered states exist, but there are many that are similarly disordered.

“From a physical point of view, this is what defines the direction of time,” says Max Lock (TU Wien). “In the past, entropy was lower; the future is where entropy is higher.” However, quantum physics encounters a problem here: the mathematician and physicist John von Neumann showed that according to the laws of quantum physics, the entropy in a quantum system cannot change at all. If you have the full information about a quantum system, the so-called ‘von Neumann entropy’ always stays the same; it is impossible to say whether time is running forward or backward, each point in time is physically as good as any other.

The Quantum Information Gap

“But this view leaves out something important,” says Tom Rivlin (TU Vienna). “In quantum physics you can never actually have full information about a system. We can choose a property of the system that we want to measure – a so-called observable. This can be, for example, the location of a particle or its speed. Quantum theory then tells us the probabilities to obtain different possible measurement results. But according to quantum theory, we can never have full information about the system.”

Even if we know the probabilities, the actual result of a specific measurement remains a surprise. This element of surprise must be included in the definition of entropy. Instead of calculating the von Neumann entropy for the complete quantum state of the entire system, you could calculate an entropy for a specific observable. The former would not change with time, but the latter might.

Shannon Entropy: A Different Measure of Disorder

This type of entropy is called ‘Shannon entropy’. It depends on the probabilities with which different possible values are measured. ‘You could say that Shannon entropy is a measure of how much information you gain from the measurement,’ says Florian Meier (TU Wien). “If there is only one possible measurement result that occurs with 100% certainty, then the Shannon entropy is zero. You won’t be surprised by the result, you won’t learn anything from it. If there are many possible values with similarly large probabilities, then the Shannon entropy is large.”

Quantum Disorder Increases After All

The research team has now been able to show that if you start with a state of low Shannon entropy, then this kind of entropy increases in a closed quantum system until it levels off around a maximum value – exactly as is known from thermodynamics in classical systems. The more time passes, the more unclear the measurement results become and the greater the surprise that can be experienced when observing. This has now been proven mathematically and also confirmed by computer simulations that describe the behavior of several interacting particles.

“This shows us that the second law of thermodynamics is also true in a quantum system that is completely isolated from its environment. You just have to ask the right questions and use a suitable definition of entropy,” says Marcus Huber.

Quantum Thermodynamics and Future Applications

If you are investigating quantum systems that consist of very few particles (for example, a hydrogen atom with only a few electrons), then such considerations are irrelevant. But today, especially with regard to modern technical applications of quantum physics, we are often faced with the challenge of describing quantum systems that consist of many particles. “To describe such many-particle systems, it is essential to reconcile quantum theory with thermodynamics,” says Marcus Huber. “That’s why we also want to use our basic research to lay the foundation for new quantum technologies.”

Reference: “Emergence of a Second Law of Thermodynamics in Isolated Quantum Systems” by Florian Meier, Tom Rivlin, Tiago Debarba, Jake Xuereb, Marcus Huber and Maximilian P.E. Lock, 14 January 2025, PRX Quantum.
DOI: 10.1103/PRXQuantum.6.010309

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