When we study quantum many-body dynamics, entanglement entropy is usually the first quantity we look at. It compresses an enormous quantum state into a single number, telling us when entanglement grows, how fast it grows, and where it eventually saturates. Over time, “increasing entanglement entropy” has almost become synonymous with “entanglement is developing.”
But is more really the whole story?
Two states can have the same total amount of entanglement while having completely different internal structures. Think of two people holding the same amount of money: one keeps almost all of it in a single account, while the other spreads it across hundreds or thousands of accounts. The total is identical, but the distributions are not.
The same is true for entanglement. Two quantum states with similar entanglement entropy can have very different Schmidt spectra. In one, a few Schmidt coefficients may still dominate. In the other, the weight may already have spread across a very large rank. Entanglement entropy sees the overall amount; the spectrum reveals how that entanglement is organized.
This changes the question we should ask.
Instead of tracking only how much entanglement has been generated, we can also ask what shape the entanglement spectrum is taking and which Schmidt-rank scale carries the dominant weight.
Once we do this, what looked like a single process—“entanglement growth”—splits into several distinct dynamical events that do not necessarily happen at the same time.
Opening Up a Number into a Spectrum
Consider a bipartition of a pure quantum state. Its Schmidt decomposition expresses the state as a collection of correlated modes, each carrying a certain weight. Sorting these weights from largest to smallest gives the Schmidt spectrum.
Entanglement entropy compresses this entire spectrum into one number. The full spectrum contains much more information: which components dominate, how uneven the distribution is, and at what rank scale the major weight resides.
To make the last feature visible, we introduce a simple notion of Schmidt-scale coordinate.
Imagine sliding a window through the ordered Schmidt spectrum. Starting at rank (r), we collect the total weight between (r) and (2r), and then ask which such window carries the largest weight. The weight of the winning window tells us how concentrated the spectrum is; its position tells us the dominant Schmidt scale.
If the spectral head remains dominant, the system is still in a low-rank regime. If a window at higher rank eventually overtakes the head, the dominant scale has undergone a shift of gears.
Nothing is physically moving through real space here. What moves is the location, in the rank-ordered spectrum, of where the dominant Schmidt weight resides.
Thermalization and MBL: Two Routes Through the Entanglement Spectrum
A disordered spin chain provides a particularly clear contrast. Neighboring spins interact, while each site experiences a random local field.
In the thermal regime, the system gradually thermalizes. Entanglement entropy grows rapidly, with an approximately linear regime. At the same time, the largest Schmidt weights progressively lose their dominance, and the dominant spectral window moves toward higher ranks.
Entanglement is not merely being generated. Its weight is being redistributed toward increasingly complex Schmidt scales.
The many-body localized (MBL) regime tells a different story.
Interactions still generate entanglement, and the entanglement entropy can continue to grow logarithmically. Higher-rank Schmidt components continue to appear as well. Yet the dominant spectral weight can remain pinned near the head of the spectrum, without undergoing the same shift toward higher rank scales.
This is something entanglement entropy alone cannot reveal.
Thermal and MBL systems can both exhibit entanglement growth, but the internal dynamics of their spectra can be fundamentally different. In particular, the generation of more entanglement and the transport of dominant weight across Schmidt scales can become decoupled.
Figure: Dynamics of different entanglement-related quantities in thermal and many-body-localized systems
One Spectrum, Four Clocks
The Schmidt-scale coordinate is only one way to look inside the spectrum. The shape of the spectrum is changing as well.
Anti-flatness measures how uneven the Schmidt weights are. It vanishes when all nonzero Schmidt weights are equal, and becomes large when a few large weights coexist with many much smaller ones.
Starting from a product state, there is initially only one nonzero Schmidt weight, so the anti-flatness is zero. As the system evolves, new Schmidt components appear, but they are highly unequal in magnitude. The spectrum therefore becomes increasingly uneven.
Later, as weight spreads across more and more components, the distribution begins to flatten again. Anti-flatness consequently develops a pronounced intermediate-time peak—a temporary barrier in the dynamics.
There is another quantity that follows its own clock: magic.
Magic is a resource-theoretic notion in quantum computation. Certain highly entangled states—stabilizer states, for example—can nevertheless be simulated efficiently on a classical computer. Nonlocal magic quantifies how far the evolving entanglement structure departs from this efficiently simulable stabilizer description.
Its dynamics can develop an intermediate-time barrier of its own.
We can therefore place four characteristic events on the same time axis:
- The entanglement clock: the peak in the entanglement-growth rate, around [figure];
- The roughness clock: the peak of anti-flatness, around [figure];
- The magic clock: the peak of the exact nonlocal magic, around [figure];
- The shift clock: the point at which more than half of the samples have left the spectral head, around [figure].
The important point is their ordering.
Even after the entanglement-growth rate has peaked, the entanglement spectrum continues to reorganize. After the spectrum reaches its maximum roughness and nonlocal magic reaches its peak, the dominant spectral weight still takes additional time to migrate toward higher Schmidt scales.
This ordering remains robust across system sizes and disorder strengths. In the MBL regime, however, the fourth clock may simply never ring.
Figure: The ordering of the four dynamical clocks
Why Does the Shift Clock Ring Last?
The answer lies in the competition between the head and the tail of the Schmidt spectrum.
Imagine that the largest Schmidt weight sits at the head, while the remaining weight forms a long tail. As weight flows from the head into the tail, different measures respond at different thresholds.
When the tail carries roughly one quarter of the total weight, the contrast between the dominant head and the rest of the spectrum is strongest, producing the peak in spectral roughness.
As the tail approaches roughly half of the total weight, the nonlocal magic reaches its maximum.
Only when the tail grows sufficiently large—roughly beyond two thirds in the relevant spectral picture—can a higher-rank window overtake the original head and trigger the shift of the dominant Schmidt scale.
In this sense, the ordering of roughness → magic → shift is already encoded in the geometry of the spectrum itself.
The same ordering also appears in random quantum circuits, suggesting that these clocks are not merely a peculiarity of one particular spin-chain model, but may reflect a more general feature of entanglement-spectrum dynamics.
What Changes Is Not Just the Quantity We Measure, but the Question We Ask
Each observable answers a different question:
- Entanglement entropy: How much entanglement is there?
- Anti-flatness: How uneven is the Schmidt spectrum?
- Nonlocal magic: How far has the entanglement structure moved away from a stabilizer description?
- Schmidt-scale coordinate: At what rank scale does the dominant weight reside?
The point of introducing four clocks is not simply to add four more curves to a plot of entanglement entropy.
It is to separate processes that have traditionally been compressed into the single phrase “entanglement growth.”
Entanglement is generated.
The spectrum becomes rough.
Nonlocal magic reaches a barrier.
And eventually, the dominant weight shifts to a new Schmidt-rank scale.
Entanglement entropy is an extraordinarily useful number. Perhaps it has become too convenient: by reducing a complicated spectrum to a single scalar, it can make several distinct dynamical processes look like one.
Once we open the spectrum back up, a hidden set of gears becomes visible.
They do not all turn together.
And the dynamics of entanglement, it turns out, is not simply a story of getting more entangled. It is also a story of reorganization, redistribution, and changing scale.
References
- Lv Zhang, Shi-Xin Zhang, Heng Fan, and Shuo Liu, Revealing Entanglement-Growth Mechanisms through the Magic Barrier, arXiv:2607.09875 (2026).
- Shi-Xin Zhang, Shuo Liu, and Yu-Qin Chen, Entanglement Growth as Transport Across Schmidt Scales, arXiv:2609.06643 (2026).
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