`It from Bit': is there a second law of quantum complexity?

In the grand story of the universe, there is a famous rule that says things tend to get messier over time. If you leave a cup of hot coffee on a table, it cools down until it matches the room temperature, and it never spontaneously reheats itself. This is the second law of thermodynamics, a principle that governs how heat and energy flow. It suggests that eventually, the universe will reach a state of perfect balance, a place where nothing changes, nothing moves, and all energy is evenly spread out. Scientists call this the "heat death." For a long time, physicists believed that once a system reached this state of maximum messiness, or entropy, the story was over. The system would just sit there, frozen in time, with no further evolution.

However, a new perspective has emerged from the intersection of gravity and quantum mechanics, the physics of the very small. While the classical laws of thermodynamics say the story ends at equilibrium, the quantum world tells a different tale. Even after a system has cooled down and reached a perfect balance, its internal quantum state continues to change. It keeps exploring new possibilities, becoming more intricate and harder to describe, long after the heat has died out. This idea, often called "life after heat death," suggests that there is a hidden layer of evolution that classical physics misses. The central question for researchers has been whether there is a new rule, a "second law of quantum complexity," that governs this slow, endless growth of intricacy.

A team of physicists at Shiraz University in Iran has now taken a significant step toward answering this question. They propose that this mysterious growth of complexity is not a random accident, but a direct consequence of how information and energy are linked. By treating the fundamental laws of motion as rules about how systems manage their information, the researchers have derived a mathematical description of how complexity grows after a system reaches equilibrium. Their work suggests that even in a universe that has seemingly stopped changing, the quantum state of matter is quietly becoming more complex, following a specific pattern that eventually slows down and settles into a maximum limit.

The researchers began by re-examining a cornerstone of physics known as the principle of least action. This principle states that when a particle moves from one point to another, it follows a path that requires the least amount of effort or "action." The team realized that this principle could be understood as a rule about information. They argued that nature prefers paths that create the least amount of new uncertainty, or "missing information." In other words, physical systems naturally try to be as efficient as possible with their information resources. This idea connects the movement of particles to the flow of information, suggesting that the laws of physics are deeply rooted in how information is organized and preserved.

Using this information-based view, the team explored what happens to a quantum system after it has reached thermal equilibrium. They applied a rule called the quasistatic information quantization rule, which links the energy of a system to the information it holds. They found that even when a system is in a state of perfect balance, the history of how it got there leaves a mark. The system continues to evolve, but not in the way classical thermodynamics predicts. Instead of staying static, the quantum state keeps growing in complexity. The researchers calculated that this complexity increases over time, following a curve that rises quickly at first and then gradually slows down as it approaches a maximum limit. This limit is determined by the total amount of disorder, or entropy, the system has.

The paper presents a specific prediction for how this growth happens. In the long run, after the system has reached equilibrium, the complexity does not grow forever at a steady pace. Instead, it follows a pattern where it increases rapidly at first, then the rate of growth slows down, and finally, it levels off at a maximum value. This behavior is different from earlier ideas that suggested complexity might grow in a straight line forever. The new model shows that the growth is actually an intermediate phase, a temporary speed-up before the system finally settles into its maximum state of complexity. This provides a more detailed and testable picture of what happens in the "afterlife" of a thermal system.

The study also looked at what happens when a system is not in equilibrium. In these cases, the researchers found that the opposite can occur. Instead of complexity growing, the system's "uncomplexity"—the amount of simple, ordered structure available for computation—can increase. This means that if a system is disturbed or prepared in a specific way, it can regain its computational resources. This finding helps explain how order can arise in the universe, such as the formation of stars and galaxies, by showing that non-equilibrium conditions can replenish the information needed to build complex structures.

Ultimately, this work connects the dots between the principle of least action, the flow of information, and the mysterious growth of quantum complexity. It suggests that the universe is not just a machine that runs down and stops, but a system that continues to explore its own possibilities long after the heat has faded. The researchers have shown that this evolution is not random but follows a predictable law derived from the fundamental relationship between energy and information. While the idea remains a conjecture that needs further testing against simulations and observations, it offers a compelling new way to understand the deep, hidden life of the quantum world. It reinforces the view that information is a fundamental building block of reality, shaping how the universe evolves from the very beginning to the very end.