On September 6, we uploaded Entanglement Growth as Transport Across Schmidt Scales to arXiv. The central idea was a simple shift in perspective: when studying entanglement dynamics, it is not enough to ask how much entanglement has grown. We should also ask which Schmidt scales carry the dominant spectral weight, and how that weight moves over time.
While working on that paper, two natural questions kept coming back to us.
Can transport across Schmidt scales be connected to a more direct and operational quantum-information task? And how does the slow diffusion of a conserved charge reshape this process?
Those questions became the starting point of our new work.
Over the following ten days, three independent papers appeared on arXiv. On September 15, arXiv:2609.16935 studied nonlocal magic and spectral structure across the many-body localization crossover. On September 16, arXiv:2609.18691 showed that chaotic dynamics can generate transient entanglement embezzlement at intermediate times. On September 17, arXiv:2609.20951 connected nonlocal magic and entanglement embezzlement to many-body dynamics, including free-fermion dynamics. All three papers cited our earlier work.
In little more than a week, Schmidt-spectrum structure, nonlocal magic, entanglement embezzlement, and many-body transport began to converge around a common nonequilibrium perspective.
A lively research direction is becoming visible.
This unusually rapid sequence of developments also says something about the changing pace of research. New tools, including AI, are shortening the cycles of literature search, numerical implementation, and cross-checking. Ideas emerging in different places can meet each other much faster than before.
The scientific questions still come from accumulated understanding, discussion, and judgment. What has changed is the feedback loop: the distance from a question to a concrete result can increasingly be measured in days rather than months.
That is the sense of urgency we felt while working on our new paper, Entanglement Embezzlement from Diffusive Hydrodynamics.
From “Where Is the Entanglement?” to “What Can It Do?”
Our previous work decomposed the Schmidt spectrum into exponentially growing rank windows and used these “doubling windows” to track whether the dominant spectral weight resides at low or high Schmidt rank.
This revealed something that the entanglement entropy alone does not show: entropy growth, spectral reshaping, and the migration of dominant weight across Schmidt scales do not necessarily happen at the same time.
The new work asks a more operational question:
Can this motion in the Schmidt spectrum actually be turned into a useful quantum-information task?
The answer is entanglement embezzlement.
Suppose Alice and Bob share a high-dimensional entangled state. Using only local operations and classical communication, they can “borrow” several Bell pairs from this state while leaving the original state globally almost unchanged. The high-dimensional entangled state therefore acts as a kind of catalyst.
How much entanglement can be borrowed depends not only on the total amount of entanglement, but also on how broadly the Schmidt weight is distributed along the logarithmic Schmidt-rank axis.
Borrowing a maximally entangled state of dimension $d$ effectively shifts the target spectrum by $\log d$ along this axis.
If the original spectrum is sufficiently broad, this shift represents only a small relative displacement, and the borrowed state can remain close to the original one.
In this way, the qualitative notion of transport across Schmidt scales from our previous paper becomes a directly quantifiable quantum-information task: one can ask how many Bell pairs can be borrowed at a specified error tolerance.
The Schmidt spectrum viewed as a distribution over logarithmic Schmidt rank, and the spectral width that controls entanglement embezzlement.
Why Can a Conserved Charge Act as an Entanglement Account?
To understand the mechanism quantitatively, consider a random pure state in a fixed global $U(1)$ charge sector. Such states provide a useful model for the equilibrium states of chaotic particle-number-conserving systems.
The total particle number is fixed, but the particle number inside a subsystem fluctuates. Different subsystem charge sectors therefore occupy different regions of the Schmidt spectrum.
Away from half filling, an effective charge bias converts ordinary particle-number fluctuations into a corresponding width on the logarithmic Schmidt-rank axis.
This gives us a finite-error conversion law: at a fixed error tolerance, the amount of entanglement that can be borrowed grows with system size according to a definite scaling law.
The result goes beyond the usual asymptotic question of whether embezzlement is possible. It gives a quantitative finite-size relation and, for arbitrary target dimension, an optimal fidelity curve.
The most striking contrast appears at half filling.
There, particle-hole symmetry makes the effective charge bias vanish. The system can still possess a maximal volume-law entanglement entropy, yet the amount of entanglement that can be borrowed goes to zero.
This cleanly separates two notions that are often treated as nearly synonymous:
having a lot of entanglement is not the same as having entanglement that is easy to borrow.
The distinction is invisible if one looks only at the entropy.
Entropy Has Already Arrived. Diffusion Is Still Catching Up.
The equilibrium analysis tells us where the system eventually ends up. The dynamical question is how it gets there.
Consider a one-dimensional $U(1)$-conserving random quantum circuit. The leading volume-law entanglement entropy develops rapidly and saturates on a ballistic timescale $t \sim L$, while complete relaxation of the conserved charge requires the much longer diffusive timescale
$$
t_{\mathrm{diff}} \sim L^2.
$$
This creates a parametrically broad intermediate regime in which the entanglement entropy already looks saturated, while the internal structure of the Schmidt spectrum is still evolving.
This is precisely where entanglement embezzlement becomes useful.
Diffusion causes the variance of the subsystem charge to grow as
$$
\mathrm{Var}(Q) \sim t^{1/2},
$$
so the standard deviation grows as
$$
\sigma_Q \sim t^{1/4}.
$$
The charge bias then maps this fluctuation width onto a width in logarithmic Schmidt rank. Under the conditions that the charge envelope is sufficiently smooth and that each charge sector is internally well mixed, the theory predicts that the amount of borrowable entanglement in the intermediate-time regime grows as
$$
E_{\mathrm{embezzle}} \sim t^{1/4}.
$$
Large-scale two-replica tensor-network simulations give a numerical slope close to this theoretical prediction, together with a consistent finite-size scaling governed by the diffusive timescale.
The $1/4$ exponent has a simple physical origin. It comes from a chain of transformations:
$$
\text{charge diffusion}
\;\longrightarrow\;
\text{charge variance}
\;\longrightarrow\;
\text{fluctuation width}
\;\longrightarrow\;
\text{Schmidt-spectrum width}
\;\longrightarrow\;
\text{borrowable entanglement}.
$$
More explicitly, diffusion produces a $t^{1/2}$ growth of the charge variance. Taking the square root gives a $t^{1/4}$ growth of the fluctuation width. That width determines how large a spectral translation can be accommodated at fixed error, and therefore how much entanglement can be borrowed.
Entropy has already arrived. Diffusion is still catching up.
Entanglement embezzlement provides a way to see this hidden late-stage dynamics that the entropy has already forgotten.
A New Phase of Research Measured in Days
The cluster of papers appearing over the past two weeks approaches the problem from different models and different physical questions, but they point toward a common object: the structure of the entanglement spectrum as a dynamical coordinate.
Schmidt scales are beginning to provide a bridge between quantum resources and nonequilibrium many-body dynamics.
Our new work takes the next step along this direction. Instead of treating motion in the Schmidt spectrum merely as a diagnostic, we ask what physical resource that motion actually makes available.
How much of the spectral structure can be turned into borrowable entanglement?
This turns a family of spectral diagnostics into an operational question in quantum information, and suggests that entanglement embezzlement can serve as a probe of dynamical structure that is invisible to conventional entanglement measures.
There is also something broader happening here.
AI and other new tools do not create scientific questions by themselves. But once an idea is already in motion, they can make it move much faster: literature can be searched more quickly, calculations can be implemented sooner, numerical results can be cross-checked almost immediately, and related ideas emerging elsewhere can be connected with much less delay.
The result is a different research tempo.
Ideas that are already on the road can now accelerate.
It really does feel faster—and there is a sense that it is urging you to keep running.
References
- Shi-Xin Zhang, Shuo Liu, and Yu-Qin Chen, “Entanglement Growth as Transport Across Schmidt Scales,” arXiv:2609.06643 (2026).
- Shan-Zhong Li and Zhi Li, “Nonlocal Magic across the Many-Body Localization Crossover,” arXiv:2609.16935 (2026).
- Matias Karjula, Teemu Ojanen, Kim Pöyhönen, Tapio Ala-Nissila, and Moein N. Ivaki, “Universal Entanglement Embezzlement and Divergent Nonlocal Magic from Generic Local Chaotic Quantum Evolution,” arXiv:2609.18691 (2026).
- Sreemayee Aditya, Piotr Sierant, and Xhek Turkeshi, “Nonlocal Magic Spreading in Many-body Quantum Dynamics: From Chaotic Evolution to Quasi-particle Picture in Integrable Models,” arXiv:2609.20951 (2026).
- Shi-Xin Zhang, Shuo Liu, and Yu-Qin Chen, “Entanglement Embezzlement from Diffusive Hydrodynamics,” arXiv:2609.26362 (2026).
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