When Does a Quantum System Forget Its Initial State?

An isolated quantum system evolves reversibly, yet a small part of it often appears to forget how the system was prepared. Understanding that apparent loss of memory is the starting point for interpreting disorder, scars, and prethermal control. The important question is operational: which observable relaxes, toward what reference ensemble, and over what accessible time? Agarwal's work is most clearly situated by distinguishing mechanisms that obstruct relaxation from measurements that merely fail to see it.

Figure to read

Disorder-dependent dynamical exponents in the original XXZ-chain study. Figure 1 in the source paper. Agarwal, Kartiek; Gopalakrishnan, Sarang; Knap, Michael; Mueller, Markus; Demler, Eugene. Anomalous diffusion and Griffiths effects near the many-body localization transition, Phys. Rev. Lett. 114, 160401 (2015). Figure linked rather than reproduced; the archived manuscript does not provide an explicit open reproduction license.

Compare the transport and relaxation exponents with the spectral diagnostic. Treat the plotted boundary as the interpretation of the finite-size 2015 study, not a settled modern phase boundary.

What thermalization means locally

For a pure state evolving under a fixed Hamiltonian, global von Neumann entropy remains unchanged. Local entropy can nevertheless increase as degrees of freedom become entangled. The eigenstate thermalization hypothesis explains why local expectation values in many energy eigenstates resemble thermodynamic predictions, subject to conserved quantities and the relevant energy window. It is a framework for local behavior, not a claim that the entire wavefunction becomes a classical thermal mixture. 1

Different tests probe different aspects of this picture. A density imbalance measures memory of an initial spatial pattern. Level statistics characterize spectral correlations. Entanglement growth tracks the spread of quantum correlations. Conductivity tests the ability to transport a conserved quantity. These measurements can have different crossover scales. Agreement among them strengthens an interpretation, while disagreement may reveal distinct mechanisms rather than a failed calculation. None should be promoted to a universal standalone definition of thermalization.

Four reasons relaxation can be slow

Integrability can supply many conservation laws, leaving stationary states constrained beyond energy and particle number. Disorder can suppress transport through interference and rare bottlenecks. High-frequency driving can produce a prethermal regime with an approximately conserved effective Hamiltonian. Quantum scars can support exceptional states or trajectories amid otherwise thermal behavior. These categories need not be experimentally easy to distinguish, but their theoretical assumptions differ. In particular, prethermal behavior explicitly allows eventual relaxation. 2

Agarwal and collaborators' Griffiths analysis addresses slow dynamics arising from rare spatial regions. Its lesson is observable-dependent: the same material can have one correlator dominated by unusual insulating segments and another controlled by conventional hydrodynamics. Conservation laws and dimensionality affect which slow process wins at late times. This is why measuring a power-law tail should prompt a mechanistic comparison rather than an immediate phase label. 3

The scar literature provides a useful contrast. The Rydberg-chain setting that motivated modern scar studies exhibits strong dependence on the chosen initial state. Special revivals coexist with more typical relaxation elsewhere in Hilbert space. Agarwal's later work on restricted operator algebras belongs to this search for structured exceptions, not to a claim that every state in a clean interacting system remains nonthermal. 4

The finite-size problem today

The current MBL debate focuses on the difference between a robust finite-time localized regime and a stable asymptotic phase. Numerical studies often show significant drift as system size changes. Resolving that drift is difficult because the Hilbert space grows exponentially while the relevant relaxation times may grow even faster. The 2025 review by Sierant and collaborators makes this distinction central and cautions against naive extrapolation from accessible scales. 5

Recent superconducting-processor work examining two-dimensional disordered dynamics provides a complementary experimental route. Its size-dependent relaxation is relevant to delocalization, but an open quantum device has a finite coherence window and calibration errors. Comparing such experiments with isolated-system theory requires identifying where external decoherence could imitate or accelerate intrinsic relaxation. The cited 2025 preprint offers size-resolved evidence over the device's accessible evolution window. 6

A practical diagnostic sequence

For a new result, first specify the conserved quantities and the initial state. Then compare local relaxation, transport, and entanglement at several sizes or observation times. Examine distributions across disorder samples, not only their averages. Finally, test whether a competing explanation—integrability, a prethermal plateau, a rare bottleneck, or environmental noise—fits the same data. Follow Griffiths physics, scars, and symmetry engineering for concrete cases. This sequence turns “failure to thermalize” into a set of falsifiable questions rather than a catch-all description of slow motion.

References

1. D'Alessio, Luca; Kafri, Yariv; Polkovnikov, Anatoli; Rigol, Marcos. From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics. Adv. Phys. 65, 239 (2016). Open manuscript.
2. Abanin, Dmitry; De Roeck, Wojciech; Ho, Wen Wei; Huveneers, Francois. A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems. Communications in Mathematical Physics 354, 809-827 (2017). Open manuscript.
3. Gopalakrishnan, Sarang; Agarwal, Kartiek; Demler, Eugene; Huse, David A.; Knap, Michael. Griffiths effects and slow dynamics in nearly many-body localized systems. Phys. Rev. B 93, 134206 (2016). Open manuscript.
4. Turner, Christopher J.; Michailidis, Alexios A.; Abanin, Dmitry A.; Serbyn, Maksym; Papic, Zlatko. Weak ergodicity breaking from quantum many-body scars. Nature Physics 14, 745–749 (2018). Open manuscript.
5. Sierant, Piotr; Lewenstein, Maciej; Scardicchio, Antonello; Vidmar, Lev; Zakrzewski, Jakub. Many-body localization in the age of classical computing
   *. Reports on Progress in Physics 88, 026502 (2025). Open manuscript.
6. Li, Tian-Ming; Sun, Zheng-Hang; Shi, Yun-Hao; Bao, Zhen-Ting; Wang, Yong-Yi; Zhang, Jia-Chi; Liu, Yu; Deng, Cheng-Lin; Yu, Yi-Han; Liu, Zheng-He; Chen, Chi-Tong; Li, Li; Li, Hao; Liu, Hao-Tian; Zhou, Si-Yun; Peng, Zhen-Yu; Liu, Yan-Jun; Wang, Ziting; Xu, Yue-Shan; Zhao, Kui; He, Yang; Feng, Da'er; Song, Jia-Cheng; Fang, Cai-Ping; Deng, Junrui; Xu, Mingyu; Chen, Yu-Tao; zhou, Bozhen; Liang, Gui-Han; Xiang, Zhong-Cheng; Xue, Guangming; Zheng, Dongning; Huang, Kaixuan; Wang, Zheng-An; Yu, Haifeng; Sierant, Piotr; Xu, Kai; Fan, Heng. Many-body delocalization with a two-dimensional 70-qubit superconducting quantum simulator. Preprint, arXiv:2507.16882 (2025); journal publication not verified as of 5 September 2026. Open manuscript.

Independent research synthesis. Literature checked 5 September 2026; preprints are identified in the references.