Nonlocal Magic Spreading in Many-body Quantum Dynamics: From Chaotic Evolution to Quasi-particle Picture in Integrable Models

In the quantum world, the behavior of many particles together is often defined by two distinct resources: entanglement and a property known as magic. Entanglement is the famous link where particles become so deeply connected that the state of one instantly influences the other, no matter the distance between them. Magic, in this specific scientific context, refers to a different kind of complexity that makes a quantum system impossible to simulate using standard classical computers. While entanglement has been studied for decades, magic is a newer concept that helps scientists understand why some quantum systems are so hard to predict. Researchers have long suspected that these two resources are linked, but they behave very differently over time. Entanglement usually grows steadily and linearly, while magic tends to rise quickly and then fade away. The big question has been whether there is a specific type of magic that is inextricably tied to entanglement, one that cannot be removed by simply changing how we look at the system. This "nonlocal magic" is the irreducible core of quantum complexity, and understanding how it spreads through a system could reveal the fundamental limits of what quantum computers can do and how nature processes information.

A team of physicists has now mapped out exactly how this nonlocal magic spreads, discovering that the answer depends entirely on whether the system is chaotic or orderly. The researchers focused on a specific measure of this magic that survives even when scientists try to simplify the system by changing the local perspective. They found that this quantity is directly controlled by the "capacity of entanglement," a concept that measures how much the energy levels of a system fluctuate. By connecting the difficult-to-calculate magic to this more manageable measure of fluctuation, the team was able to predict the behavior of quantum systems ranging from chaotic chaos to perfectly ordered motion. Their work, supported by large-scale computer simulations, reveals a sharp divide in how nature handles complexity. In chaotic systems, which scramble information rapidly, nonlocal magic builds up quickly but is temporary. It rises, hits a peak, and then decays to a small, constant value. This means that the most chaotic systems, which are the best at scrambling information, actually lose their ability to sustain this deep form of complexity over time.

In contrast, the researchers found that orderly systems, specifically those made of free-moving particles that do not interact with each other, behave very differently. In these integrable models, the nonlocal magic also grows at the start, but instead of fading away, it settles into a steady state that keeps growing as the system gets larger. The team developed a picture of these systems as streams of independent particles, or quasiparticles, that carry the magic from one part of the system to another. Because these particles do not collide or scramble, the fluctuations in the system's energy remain strong enough to sustain the magic indefinitely. The simulations showed that for these free-fermion systems, the amount of nonlocal magic grows logarithmically with the size of the system, meaning it never stops increasing as you add more particles. This suggests that while chaotic systems are excellent at mixing information, they are actually poor at maintaining the specific type of complexity required for certain advanced quantum tasks.

The study also explored systems that sit somewhere in between, such as those with energy conservation but no perfect order. In these cases, the magic still rises and falls, but the decay is much slower than in fully chaotic systems. The researchers confirmed that the key to whether magic lasts or disappears lies in the shape of the system's energy spectrum. If the spectrum becomes flat, the magic vanishes; if the spectrum retains enough variation, the magic persists. This finding has a surprising implication for a theoretical concept called "entanglement embezzlement," which describes the ability to extract entanglement from a system without disturbing it. The results show that the strongest scramblers, which one might expect to be the most powerful, can only embezzle entanglement for a short time. Conversely, the simpler, free-fermion systems, which are easier to simulate, can sustain this ability forever. The work demonstrates that the capacity to hold onto complex quantum resources is not about how much scrambling occurs, but about the specific structure of the fluctuations that remain after the system settles down.

The researchers tested these ideas using two main types of simulations. First, they looked at random quantum circuits and chaotic Hamiltonians, which represent the most disordered, scrambling environments. In these scenarios, the nonlocal magic followed a rise-peak-fall pattern. It grew logarithmically with time, reached a maximum that depended on the size of the system, and then decayed exponentially to a value that did not depend on the system size at all. This confirmed that in chaotic environments, the irreducible magic is a transient phenomenon. Second, they examined free-fermionic systems, where particles move without interacting. Here, the magic grew logarithmically and then saturated at a value that continued to increase as the system size increased. The team used a quasiparticle model to explain this, showing that the particles carry the necessary fluctuations to the subsystem, keeping the magic alive. They also extended this analysis to interacting integrable models, where particles do interact but in a highly ordered way, and found that the same principle held: the magic persisted as long as the system retained specific spectral fluctuations.

One of the most significant aspects of this work is the method used to reach these conclusions. Calculating nonlocal magic directly is extremely difficult because it requires optimizing over all possible local changes of basis, a task that becomes impossible for large systems. The authors bypassed this obstacle by relating the magic to the capacity of entanglement, a quantity that can be calculated from just the first two statistical moments of the energy spectrum. This connection allowed them to make precise predictions without needing to reconstruct the entire energy spectrum. Their simulations showed that this approximation becomes increasingly accurate as the system size grows, particularly in the free-fermion models where the distribution of energy levels becomes very smooth. The results were consistent across different types of dynamics, including random circuits, chaotic Hamiltonians, and free-fermion chains, providing a unified framework for understanding how quantum complexity evolves.

The findings challenge the intuitive notion that more scrambling leads to more powerful quantum resources. Instead, the study suggests that the strongest scramblers are actually too efficient at flattening the energy landscape, which destroys the very fluctuations needed to sustain nonlocal magic. In contrast, systems that are less chaotic, like free-fermion chains, preserve these fluctuations and can maintain a high level of nonlocal magic indefinitely. This distinction has operational consequences for quantum information processing. It implies that for tasks requiring the sustained extraction of entanglement, highly chaotic systems may not be the best choice, while more orderly systems could offer a stable resource. The researchers also noted that their framework could be applied to other areas, such as systems with measurement or dissipation, though these remain open questions for future investigation.

Ultimately, this paper provides a clear map of how nonlocal magic spreads and survives in different quantum environments. It establishes that the fate of this resource is determined by the capacity of entanglement, a measure of spectral fluctuations. In chaotic systems, this capacity is transient, leading to a temporary burst of magic that eventually fades. In integrable systems, the capacity remains extensive, allowing the magic to grow and persist. The work bridges the gap between chaotic and ordered dynamics, showing that the ability to sustain quantum complexity is not a matter of degree but of kind. By focusing on the shape of the energy spectrum rather than just the amount of entanglement, the researchers have uncovered a fundamental principle governing the life cycle of quantum resources. This insight not only deepens our theoretical understanding of many-body physics but also offers practical guidance for designing quantum systems that can maintain their complex properties over time.