Pith. sign in

REVIEW 2 major objections 6 minor 126 references

Revealing Entanglement-Growth Mechanisms through the Magic Barrier

T0 review · 2 major / 6 minor · reviewed 2026-07-14 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The relative timing of the magic barrier and peak entropy growth diagnoses whether bipartite entanglement is built locally or mainly transported.

desk verdict Clean, usable diagnostic: peak separation between anti-flatness and entropy growth tracks build vs transport, with solid XXZ and circuit evidence beyond the thermal correlation in [83]. read the letter →

arxiv 2607.09875 v1 pith:U3PTO2A2 submitted 2026-07-10 quant-ph

Lv Zhang , Shi-Xin Zhang , Heng Fan , Shuo Liu This is my paper · ORCID
classification quant-ph
keywords magicbarrieranti-flatnessentanglementgrowthmany-bodylocalizationquantumSchmidtspectrumrandom-fieldXXZbuildversustransport
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Entanglement and magic are complementary quantum resources, but how they evolve together in many-body dynamics has been unclear. This paper argues that the mechanism of bipartite entanglement growth is encoded in a simple relative timescale: the delay between the peak of the entropy growth rate and the transient peak of entanglement-spectrum anti-flatness (the magic barrier). When entanglement is generated locally across a cut, the same process both raises entropy and roughens the Schmidt spectrum, so the two peaks coincide. When entanglement is mainly redistributed or transported, entropy can rise before spectral non-flatness develops, and the peaks separate. The authors show this separation grows across the thermal-to-MBL crossover of a disordered XXZ chain, grows further when the initial state already stores short-range entanglement, and shrinks when a random circuit is tuned to favor local Haar gates over pure SWAP transport. The result turns the magic barrier into a spectral diagnostic of how quantum information is generated, moved, and reshaped.

What carries the argument

The magic barrier: the transient peak of anti-flatness F_A of the entanglement spectrum (the variance of Schmidt eigenvalues sampled with probability equal to themselves). Comparing its time t*_F with the time t*_Ṡ of maximal entropy growth yields the separation that diagnoses build versus transport.

What would settle it

In a system known to be transport-dominated (or pure SWAP of flat Bell pairs), measure both peaks and check whether their separation remains large; if the anti-flatness peak still coincides with the entropy-growth peak, the claimed diagnostic fails.

Watch

Extended reading notes

Core claim

The mechanism of bipartite entanglement growth is encoded in the relative timescale between the entropy-growth-rate peak and the magic barrier (the transient peak of entanglement-spectrum anti-flatness). Local build keeps the peaks in the same window; transport or redistribution separates them. This is shown in the random-field XXZ chain across the thermal-MBL crossover and confirmed with Bell-pair initial states and a tunable SWAP-Haar circuit.

Load-bearing premise

That in the localized regime entropy can grow by distance-dependent dephasing or redistribution of pre-existing blocks without immediately roughening the dominant Schmidt weights, so the first entropy-growth peak systematically precedes the first anti-flatness peak.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The manuscript argues that the mechanism of bipartite entanglement growth is encoded in the relative timescale Δt_sep = t*_F − t*_Ṡ between the transient peak of entanglement-spectrum anti-flatness F_A (the “magic barrier”) and the peak of the entropy growth rate Ṡ_A. Local build processes that expand and reshape Schmidt weights keep the two peaks correlated; transport or redistribution of pre-existing entanglement can increase S_A before appreciable spectral non-flatness develops, separating the peaks. The claim is tested in the random-field XXZ chain across the thermal–MBL crossover (product and Bell-pair initial states), supported by Schmidt-level build/transport algebra in the SM, and benchmarked in a tunable SWAP–Haar circuit where the Haar fraction r continuously reduces Δt_sep.

Significance. If the diagnostic holds, it supplies a concrete spectral probe of how bipartite entanglement is generated versus redistributed, linking two complementary resources (entanglement and magic/anti-flatness) at the level of dynamical timescales rather than static resource measures. Strengths include: (i) independent definitions of t*_F and t*_Ṡ from F_A(t) and S_A(t); (ii) elementary Schmidt algebra for build (source term R_3−R_2^{2} = Var_x) and pure Bell transport (F_A ≡ 0); (iii) a controlled circuit that interpolates the build fraction; (iv) a same-Hamiltonian Bell-pair stress test; and (v) finite-size Krylov checks of Δt_sep for L = 16–22. Relative to prior observations of correlated peaks in thermal settings and to build/transport language in the entanglement literature, the systematic separation across the thermal–MBL crossover and the circuit interpolation are new and falsifiable.

major comments (2)
  1. SM §III.A and Fig. S2: near the thermal–MBL crossover the authors note that later-time features of F_A(t) can become comparable to the first barrier and therefore adopt a first-local-maximum convention for t*_F and t*_Ṡ. This choice is load-bearing for the strong-disorder branch of Fig. 2(e). The main text should state the convention explicitly (one sentence is enough for a Letter) and report a brief robustness check—e.g., whether Δt_sep remains positive and systematically growing if the global maximum of F_A is used, or if a late-time window is excluded—so that the MBL-side trend is not convention-dependent.
  2. Main text “Two mechanisms…” and SM §I.D: the identification of MBL entropy growth with transport-like spectral dynamics is phenomenological (l-bit dephasing, range-dependent clocks). The central diagnostic claim does not require a microscopic proof of that identification, because the Bell-pair XXZ test and the SWAP–Haar circuit already separate build from redistribution. Still, the wording “the localized regime is transport-like in the spectral sense” should be more carefully caveated as an analogy for the relative clocks of Ṡ_A and F_A, not as a claim that MBL is equivalent to SWAP transport, to avoid over-reading Fig. 2(e) at large W.
minor comments (6)
  1. Eq. (5) and Fig. 2(e): state clearly whether peak times are extracted from ensemble-averaged traces (as SM §III.A indicates) or as averages of per-sample peak times; the two procedures can differ when peaks are broad.
  2. Fig. 1: the schematic is helpful; labeling the cut and the Schmidt-block structure more explicitly (flat block vs nonuniform split) would make the build/transport contrast easier to read at a glance.
  3. Fig. 3(d) and SM Fig. S7: report error bars or the number of trajectories used for Δt_sep(r) in the main-text caption (N_traj = 50000 is only in the SM).
  4. Introduction: the connection of F_A to nonlocal magic is cited via lower bounds; a single clarifying phrase that the Letter uses anti-flatness as a spectral diagnostic (not a full magic monotone) would prevent over-interpretation of the “magic barrier” name.
  5. SM Eq. (S16) and the p ≠ 1/2 dimer discussion are useful; a one-line pointer in the main text that pure transport of non-flat dimers can still separate flux-controlled Ṡ_A from accumulation-controlled F_A would strengthen the analytic narrative without extra figures.
  6. Typographical consistency: “magic-barrier time t*_F” vs “magic barrier peak”; fix occasional missing spaces around t^* notation in the compiled text.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: independently defined peaks, Schmidt-algebra derivation of build/transport, and new external numerics (XXZ + SWAP–Haar) support the diagnostic; one non-load-bearing self-citation to overlapping-author mechanism language.

  1. self citation load bearing [Main text p. 3, “Two mechanisms…” paragraph; also SM §I intro]
    "We provide an analytical understanding of this behavior in terms of two distinct mechanisms of entanglement growth [89], as illustrated in Fig. 1. ... A related transport-based interpretation was proposed from a complementary perspective in Ref. [89]."

    Ref. [89] shares an author (S.-X. Zhang). The citation supplies the build/transport nomenclature that organizes the interpretation. However it is not load-bearing: the paper supplies its own Schmidt-sector algebra (SM Eqs. S6–S15) and independent numerical controls (Bell-pair XXZ, tunable r circuit) that establish the peak-separation diagnostic without relying on the prior paper’s results.

full rationale

The two peak times t*_Ṡ and t*_F are extracted independently from the ensemble-averaged traces of Ṡ_A(t) and F_A(t) (first local maxima, SM §III.A); Δt_sep is a measured difference, not a fitted free parameter that is then re-predicted. The elementary build algebra (SM §I.B: F'_A = R_3 F_A + (R_3 − R_2^{2})P_2^{2} with R_3 − R_2^{2} = Var_x(x_μ)) and pure-transport limit (Bell pairs give F_A ≡ 0, SM §I.C) are self-contained and do not reduce to the target claim by construction. The XXZ thermal–MBL scan, Bell-pair initial-state stress test, and continuous r-interpolation in the SWAP–Haar circuit are new external tests that falsifiably vary the build/transport balance. The sole self-citation of note is Ref. [89] (overlapping author S.-X. Zhang) for the “two mechanisms of entanglement growth” language; the paper re-derives the spectral consequences itself and does not rest the central diagnostic claim solely on that citation. No uniqueness theorem, no ansatz smuggled via prior work, and no renaming of a known empirical pattern as a first-principles result. Score 1 reflects only the minor, non-load-bearing self-citation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The claim rests on standard spectral definitions, literature bounds linking anti-flatness to nonlocal magic, the phenomenological l-bit picture of MBL, and the build/transport dichotomy. No continuous free parameters are fitted to manufacture the peak separation; W and r are scanned control knobs. The main modeling choices are the first-peak convention and the identification of MBL dephasing with transport-like spectral behavior.

free parameters (2)
  • First-local-maximum peak convention for t*_F and t*_Ṡ
    Near the thermal–MBL crossover later F_A features can rival the first barrier; the paper restricts analysis to the first local maxima of ensemble-averaged traces (SM §III.A). This choice affects reported Δt_sep but is not a continuous fit.
  • Disorder crossover reference W ≃ 6.2
    Used as a literature anchor for the finite-size thermal–MBL crossover (Morningstar et al.); not fitted to the peak-separation data, but frames the interpretation of the scan.
assumptions (4)
  • domain assumption Anti-flatness F_A = P_3 − P_2² is the variance of Schmidt eigenvalues sampled with probability λ_α and lower-bounds nonlocal magic.
    Invoked from prior resource-theory literature (Tirrito et al., Ebner et al., etc.) to motivate calling the F_A peak a magic barrier; the dynamical diagnostic itself only needs F_A as a spectral-shape observable.
  • domain assumption MBL dynamics is described by l-bits with exponentially decaying interactions, so entanglement grows by slow distance-dependent dephasing rather than rapid local thermal scrambling of Schmidt weights.
    Used in main text and SM §I.D to explain why localized dynamics is transport-like in the spectral sense and why Δt_sep grows with W.
  • domain assumption Bipartite entanglement growth decomposes into local build (nonuniform Schmidt splitting) versus transport/redistribution of pre-existing entanglement.
    Adopted from Zhang, Li, Zhang (Phys. Rev. Lett. 2026) and made quantitative via the source–dilution identity for log F_A in SM §I.
  • standard math Standard Haar moment formulas for reduced density matrices of random pure states.
    Used in SM §II to give the late-time/thermal anti-flatness baseline that vanishes with Hilbert-space dimension.
invented entities (1)
  • Magic barrier (as operational peak time t*_F of anti-flatness) independent evidence
    purpose: Names the transient maximum of F_A so that its relative timing to t*_Ṡ can be used as a diagnostic of entanglement-growth mechanism.
    The term is taken from prior thermalization work and operationalized here as an extractable peak; it is a named observable feature, not a new physical particle or force. Independent handle is the measurable F_A(t) itself.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Revealing Entanglement-Growth Mechanisms through the Magic Barrier." pith.science (2026). https://pith.science/paper/U3PTO2A2

@misc{pith2026260709875,
  author       = {Pith},
  title        = {Pith review of: Revealing Entanglement-Growth Mechanisms through the Magic Barrier},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U3PTO2A2}},
  note         = {Machine review of arXiv:2607.09875}
}
read the original abstract

Quantum entanglement and magic are complementary resources underlying quantum computational advantage, yet their dynamical relation in many-body systems remains poorly understood. In this Letter, we show that the mechanism of bipartite entanglement growth is encoded in the relative timescale between the entropy-growth-rate peak and the magic barrier, defined as the transient peak of the anti-flatness of the entanglement spectrum. When entanglement is locally built, the same microscopic process increases the entropy and reshapes the Schmidt spectrum, so the magic-barrier peak occurs in the time window of maximal entropy growth. When entanglement is mainly transported or redistributed, entropy can grow before appreciable spectral non-flatness is generated, naturally separating the two peak times. We demonstrate this distinction in the random-field XXZ chain: the two peaks remain strongly correlated in the thermal regime, while their separation grows systematically across the thermal--MBL crossover. We further validate this theoretical framework by employing Bell-pair initial states alongside a tunable SWAP--Haar random circuit. Our results reveal an intrinsic dynamical connection between entanglement and magic, establishing the magic barrier as a powerful spectral diagnostic of how quantum information is generated, transported, and reshaped.

Figures

Figures reproduced from arXiv: 2607.09875 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of build- and transport-dominated entan [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Dynamics of anti-flatness and entropy growth in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Random-circuit benchmark initialized from Bell [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

126 extracted references · 12 linked inside Pith

  1. [83]

    C. Cao, G. Cheng, A. Hamma, L. Leone, W. Munizzi, and S. F. Oliviero, Gravitational backreaction is magi- cal, PRX Quantum6, 040375 (2025)

    2025

  2. [89]

    Morningstar, L

    A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Avalanches and many-body res- onances in many-body localized systems, Phys. Rev. B 105, 174205 (2022)

    2022

  3. [1]

    Vertical dashed lines mark the corresponding peak positionst ∗ F andt ∗ ˙S

    The upper row uses random product initial states in the half-filling sector, while the middle row uses Bell-pair initial states with no Bell pair crossing either half-chain bipartition boundary. Vertical dashed lines mark the corresponding peak positionst ∗ F andt ∗ ˙S. All data are averaged over at least 1500 disorder realizations. (e) Peak separation ∆t...

  4. [2]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quan- tum thermalization through entanglement in an isolated many-body system, Science353, 794 (2016)

    2016

  5. [3]

    Calabrese and J

    P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, Journal of Statisti- cal Mechanics: Theory and Experiment2005, P04010 (2005)

    2005

  6. [4]

    Kim and D

    H. Kim and D. A. Huse, Ballistic spreading of entan- glement in a diffusive nonintegrable system, Phys. Rev. Lett.111, 127205 (2013)

    2013

  7. [5]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X7, 031016 (2017)

    2017

  8. [6]

    Eisert, M

    J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many- body systems out of equilibrium, Nature Physics11, 124 (2015)

    2015

Show all 126 references
  1. [7]

    Calabrese and J

    P. Calabrese and J. Cardy, Quantum quenches in ex- tended systems, Journal of Statistical Mechanics: The- ory and Experiment2007, P06008 (2007)

    2007

  2. [8]

    Mezei and D

    M. Mezei and D. Stanford, On entanglement spread- ing in chaotic systems, Journal of High Energy Physics 2017, 65 (2017)

    2017

  3. [9]

    Z. Gong, A. Nahum, and L. Piroli, Coarse-grained en- tanglement and operator growth in anomalous dynam- ics, Phys. Rev. Lett.128, 080602 (2022)

    2022

  4. [10]

    P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev.109, 1492 (1958)

    1958

  5. [11]

    Basko, I

    D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Annals of Physics 321, 1126 (2006)

    2006

  6. [12]

    Oganesyan and D

    V. Oganesyan and D. A. Huse, Localization of inter- acting fermions at high temperature, Phys. Rev. B75, 155111 (2007)

    2007

  7. [13]

    Pal and D

    A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B82, 174411 (2010)

    2010

  8. [14]

    D. A. Huse, R. Nandkishore, and V. Oganesyan, Phenomenology of fully many-body-localized systems, Phys. Rev. B90, 174202 (2014)

    2014

  9. [15]

    Serbyn, Z

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, Local conserva- tion laws and the structure of the many-body localized states, Phys. Rev. Lett.111, 127201 (2013)

    2013

  10. [16]

    V. Ros, M. M¨ uller, and A. Scardicchio, Integrals of mo- tion in the many-body localized phase, Nuclear Physics B891, 420 (2015)

    2015

  11. [17]

    J. Z. Imbrie, On many-body localization for quantum spin chains, Journal of Statistical Physics163, 998 (2016)

    2016

  12. [18]

    A. C. Potter, R. Vasseur, and S. A. Parameswaran, Universal properties of many-body delocalization tran- sitions, Phys. Rev. X5, 031033 (2015)

    2015

  13. [19]

    D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field heisenberg chain, Phys. Rev. B91, 081103(R) (2015)

    2015

  14. [20]

    Ponte, Z

    P. Ponte, Z. Papi´ c, F. m. c. Huveneers, and D. A. Abanin, Many-body localization in periodically driven systems, Phys. Rev. Lett.114, 140401 (2015)

    2015

  15. [21]

    D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, Localization-protected quantum or- der, Phys. Rev. B88, 014206 (2013)

    2013

  16. [22]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys.91, 021001 (2019). 6

    2019

  17. [23]

    Sierant, M

    P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing, Reports on Progress in Physics 88, 026502 (2025)

    2025

  18. [24]

    Sierant and J

    P. Sierant and J. Zakrzewski, Challenges to observation of many-body localization, Phys. Rev. B105, 224203 (2022)

    2022

  19. [25]

    Zhang and H

    S.-X. Zhang and H. Yao, Strong and weak many-body localizations, arXiv:1906.00971 (2019)

    1906 arXiv

  20. [26]

    A. S. Aramthottil, P. Sierant, M. Lewenstein, and J. Za- krzewski, Phenomenology of many-body localization in bond-disordered spin chains, Phys. Rev. Lett.133, 196302 (2024)

    2024

  21. [27]

    P. R. N. Falc˜ ao, A. S. Aramthottil, P. Sierant, and J. Za- krzewski, Many-body localization crossover is sharper in a quasiperiodic potential, Phys. Rev. B110, 184209 (2024)

    2024

  22. [28]

    Bauer and C

    B. Bauer and C. Nayak, Area laws in a many-body lo- calized state and its implications for topological order, Journal of Statistical Mechanics: Theory and Experi- ment2013, P09005 (2013)

    2013

  23. [29]

    J. A. Kj¨ all, J. H. Bardarson, and F. Pollmann, Many- body localization in a disordered quantum ising chain, Phys. Rev. Lett.113, 107204 (2014)

    2014

  24. [30]

    Jian and H

    S.-K. Jian and H. Yao, Solvable sachdev-ye-kitaev mod- els in higher dimensions: From diffusion to many-body localization, Phys. Rev. Lett.119, 206602 (2017)

    2017

  25. [31]

    Zhang and H

    S.-X. Zhang and H. Yao, Universal properties of many- body localization transitions in quasiperiodic systems, Phys. Rev. Lett.121, 206601 (2018)

    2018

  26. [32]

    Liu, S.-X

    S. Liu, S.-X. Zhang, C.-Y. Hsieh, S. Zhang, and H. Yao, Probing many-body localization by excited-state varia- tional quantum eigensolver, Phys. Rev. B107, 024204 (2023)

    2023

  27. [33]

    Liu, H.-K

    S. Liu, H.-K. Zhang, S. Yin, S.-X. Zhang, and H. Yao, Symmetry restoration and quantum mpemba effect in many-body localization systems, Science Bulletin70, 3991 (2025)

    2025

  28. [34]

    J. H. Bardarson, F. Pollmann, and J. E. Moore, Un- bounded growth of entanglement in models of many- body localization, Phys. Rev. Lett.109, 017202 (2012)

    2012

  29. [35]

    Serbyn, Z

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, Universal slow growth of entanglement in interacting strongly disor- dered systems, Phys. Rev. Lett.110, 260601 (2013)

    2013

  30. [36]

    D. J. Luitz and Y. Bar Lev, Information propagation in isolated quantum systems, Phys. Rev. B96, 020406(R) (2017)

    2017

  31. [37]

    D.-L. Deng, X. Li, J. H. Pixley, Y.-L. Wu, and S. Das Sarma, Logarithmic entanglement lightcone in many-body localized systems, Phys. Rev. B95, 024202 (2017)

    2017

  32. [38]

    MacCormack, M

    I. MacCormack, M. T. Tan, J. Kudler-Flam, and S. Ryu, Operator and entanglement growth in nonthermalizing systems: Many-body localization and the random sin- glet phase, Phys. Rev. B104, 214202 (2021)

    2021

  33. [39]

    Xu and P

    C. Xu and P. Zhang, Entanglement growth from struc- tured initial states in many-body localized systems, arXiv:2605.20656 (2026)

    2026 arXiv

  34. [40]

    Vidal, Efficient classical simulation of slightly en- tangled quantum computations, Phys

    G. Vidal, Efficient classical simulation of slightly en- tangled quantum computations, Phys. Rev. Lett.91, 147902 (2003)

    2003

  35. [41]

    Jozsa and N

    R. Jozsa and N. Linden, On the role of entanglement in quantum-computational speed-up, Proceedings of the Royal Society A: Mathematical, Physical and Engineer- ing Sciences459, 2011 (2003)

    2011

  36. [42]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett.86, 5188 (2001)

    2001

  37. [43]

    H. J. Briegel and R. Raussendorf, Persistent entangle- ment in arrays of interacting particles, Phys. Rev. Lett. 86, 910 (2001)

    2001

  38. [44]

    Gottesman, The heisenberg representation of quan- tum computers, arXiv:quant-ph/9807006 (1998)

    D. Gottesman, The heisenberg representation of quan- tum computers, arXiv:quant-ph/9807006 (1998)

    1998 arXiv

  39. [45]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A70, 052328 (2004)

    2004

  40. [46]

    Dehaene and B

    J. Dehaene and B. De Moor, Clifford group, stabilizer states, and linear and quadratic operations over gf(2), Phys. Rev. A68, 042318 (2003)

    2003

  41. [47]

    Anders and H

    S. Anders and H. J. Briegel, Fast simulation of stabilizer circuits using a graph-state representation, Phys. Rev. A73, 022334 (2006)

    2006

  42. [48]

    T´ oth and O

    G. T´ oth and O. G¨ uhne, Entanglement detection in the stabilizer formalism, Phys. Rev. A72, 022340 (2005)

    2005

  43. [49]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Phys. Rev. A71, 022316 (2005)

    2005

  44. [50]

    Howard, J

    M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the ‘magic’ for quantum compu- tation, Nature510, 351 (2014)

    2014

  45. [51]

    Veitch, S

    V. Veitch, S. A. Hamed Mousavian, D. Gottesman, and J. Emerson, The resource theory of stabilizer quantum computation, New Journal of Physics16, 013009 (2014)

    2014

  46. [52]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer r´ enyi entropy, Phys. Rev. Lett.128, 050402 (2022)

    2022

  47. [53]

    Haug and L

    T. Haug and L. Piroli, Stabilizer entropies and nonsta- bilizerness monotones, Quantum7, 1092 (2023)

    2023

  48. [54]

    Liu and A

    Z.-W. Liu and A. Winter, Many-body quantum magic, PRX Quantum3, 020333 (2022)

    2022

  49. [55]

    Wei and Z.-W

    F. Wei and Z.-W. Liu, Long-range nonstabilizer- ness and quantum codes, phases, and complexity, arXiv:2503.04566 (2026)

    2026

  50. [56]

    P. S. Tarabunga, E. Tirrito, T. Chanda, and M. Dal- monte, Many-body magic via pauli-markov chains— from criticality to gauge theories, PRX Quantum4, 040317 (2023)

    2023

  51. [57]

    C. D. White, C. Cao, and B. Swingle, Conformal field theories are magical, Phys. Rev. B103, 075145 (2021)

    2021

  52. [58]

    P. S. Tarabunga, Critical behaviors of non-stabilizerness in quantum spin chains, Quantum8, 1413 (2024)

    2024

  53. [59]

    Y.-M. Ding, Z. Wang, and Z. Yan, Evaluating many- body stabilizer r´ enyi entropy by sampling reduced pauli strings: Singularities, volume law, and nonlocal magic, PRX Quantum6, 030328 (2025)

    2025

  54. [60]

    Hoshino, M

    M. Hoshino, M. Oshikawa, and Y. Ashida, Stabilizer r´ enyi entropy and conformal field theory, Phys. Rev. X 16, 011037 (2026)

    2026

  55. [61]

    Zhang, I

    Y. Zhang, I. H. Kim, Y. Bao, and S. Vijay, Exten- sive long-range magic in non-abelian topological orders, arXiv:2605.15150 (2026)

    2026 arXiv

  56. [62]

    Liu and B

    Z. Liu and B. K. Clark, Nonequilibrium quantum monte carlo algorithm for stabilizer r´ enyi entropy in spin sys- tems, Phys. Rev. B111, 085144 (2025)

    2025

  57. [63]

    Torre, F

    G. Torre, F. Franchini, and S. M. Giampaolo, Non-local magic from the entanglement spectrum, arXiv:2607.07808 (2026)

    2026 arXiv

  58. [64]

    J. Sun, L. Cheng, and S.-X. Zhang, Stabilizer ground states for simulating quantum many-body physics: the- ory, algorithms, and applications, Quantum9, 1782 (2025). 7

    2025

  59. [65]

    C. E. P. Robin, Stabilizer-accelerated quantum many- body ground-state estimation, Phys. Rev. A112, 052408 (2025)

    2025

  60. [66]

    X. Qian, J. Huang, and M. Qin, Augmenting den- sity matrix renormalization group with clifford circuits, Phys. Rev. Lett.133, 190402 (2024)

    2024

  61. [67]

    X. Qian, J. Huang, and M. Qin, Clifford circuits augmented time-dependent variational principle, Phys. Rev. Lett.134, 150404 (2025)

    2025

  62. [68]

    C. Fan, X. Qian, H.-C. Zhang, R.-Z. Huang, M. Qin, and T. Xiang, Disentangling critical quantum spin chains with clifford circuits, Phys. Rev. B111, 085121 (2025)

    2025

  63. [69]

    Turkeshi, E

    X. Turkeshi, E. Tirrito, and P. Sierant, Magic spreading in random quantum circuits, Nature Communications 16, 2575 (2025)

    2025

  64. [70]

    Aditya, X

    S. Aditya, X. Turkeshi, and P. Sierant, Growth and spreading of quantum resources under random circuit dynamics, Phys. Rev. Res. (2026)

    2026

  65. [71]

    Zhang and Y

    Y. Zhang and Y. Gu, Quantum magic dynamics in ran- dom circuits, npj Quantum Information12, 87 (2026)

    2026

  66. [72]

    Xiao and S

    Z. Xiao and S. Ryu, Diffusive dynamics of nonstabiliz- erness, arXiv:2606.13606 (2026)

    2026 arXiv

  67. [73]

    Xiao and S

    Z. Xiao and S. Ryu, Exponentially accelerated sampling of pauli strings for nonstabilizerness, arXiv:2601.00761 (2026)

    2026 arXiv

  68. [74]

    Huang, H.-Z

    X. Huang, H.-Z. Li, C. H. Lee, and J.-X. Zhong, A fast and exact approach for stabilizer r´ enyi entropy via the xor-fwht algorithm, arXiv:2512.24685 (2026)

    2026

  69. [75]

    Sierant, J

    P. Sierant, J. Vall` es-Muns, and A. Garcia-Saez, Com- puting quantum magic of state vectors, Quantum10, 2059 (2026)

  70. [76]

    X. Feng, C. Liu, Z. Cheng, W. W. Ho, and M. Ip- politi, Quantum resource localizability transitions in deep thermalization, arXiv:2606.08756 (2026)

    2026 arXiv

  71. [77]

    Rattacaso, L

    D. Rattacaso, L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer entropy dynamics after a quan- tum quench, Phys. Rev. A108, 042407 (2023)

    2023

  72. [78]

    Odavi´ c, M

    J. Odavi´ c, M. Viscardi, and A. Hamma, Stabilizer en- tropy in nonintegrable quantum evolutions, Phys. Rev. B112, 104301 (2025)

    2025

  73. [79]

    Tirrito, X

    E. Tirrito, X. Turkeshi, and P. Sierant, Anticoncen- tration and nonstabilizerness spreading under ergodic quantum dynamics, Phys. Rev. Lett.135, 220401 (2025)

    2025

  74. [80]

    P. R. N. Falc˜ ao, P. Sierant, J. Zakrzewski, and E. Tir- rito, Nonstabilizerness dynamics in many-body localized systems, Phys. Rev. Lett.135, 240404 (2025)

    2025

  75. [81]

    Li, Y.-R

    H.-Z. Li, Y.-R. Zhang, Y.-J. Zhao, X. Huang, and J.- X. Zhong, Slow growth of quantum nonstabilizerness in disorder-free stark many-body localization, Phys. Rev. B113, 104305 (2026)

    2026

  76. [82]

    Tirrito, P

    E. Tirrito, P. S. Tarabunga, G. Lami, T. Chanda, L. Leone, S. F. E. Oliviero, M. Dalmonte, M. Collura, and A. Hamma, Quantifying nonstabilizerness through entanglement spectrum flatness, Phys. Rev. A109, L040401 (2024)

    2024

  77. [84]

    Ebner, B

    L. Ebner, B. M¨ uller, A. Sch¨ afer, L. Schmotzer, C. Seidl, and X. Yao, Magic barrier before thermalization, Phys. Rev. Lett.136, 230403 (2026)

    2026

  78. [85]

    C. E. P. Robin and M. J. Savage, Antiflatness and non- local nonstabilizerness in two-particle scattering pro- cesses, Phys. Rev. D114, 014007 (2026)

    2026

  79. [86]

    Jasser, D

    B. Jasser, D. Iannotti, and A. Hamma, A journey through flatland: What does the antiflatness of a spec- trum teach us?, arXiv:2605.21664 (2026)

    2026 arXiv

  80. [87]

    Turkeshi, M

    X. Turkeshi, M. Schir` o, and P. Sierant, Measuring non- stabilizerness via multifractal flatness, Phys. Rev. A 108, 042408 (2023)

    2023

  81. [88]

    ˇZnidariˇ c, T

    M. ˇZnidariˇ c, T. c. v. Prosen, and P. Prelovˇ sek, Many- body localization in the heisenbergxxzmagnet in a random field, Phys. Rev. B77, 064426 (2008)

    2008

  82. [90]

    Zhang, Z.-X

    C.-Y. Zhang, Z.-X. Li, and S.-X. Zhang, Entanglement growth from entangled states: A unified perspective on entanglement generation and transport, Phys. Rev. Lett.137, 020404 (2026)

    2026

  83. [91]

    See the Supplemental Material for more details

  84. [92]

    Chandran, I

    A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Constructing local integrals of motion in the many-body localized phase, Phys. Rev. B91, 085425 (2015)

    2015

  85. [93]

    T. E. O’Brien, D. A. Abanin, G. Vidal, and Z. Papi´ c, Explicit construction of local conserved operators in dis- ordered many-body systems, Phys. Rev. B94, 144208 (2016)

    2016

  86. [94]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Probing many- body dynamics on a 51-atom quantum simulator, Na- ture551, 579 (2017)

    2017

  87. [95]

    C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Ser- byn, and Z. Papi´ c, Weak ergodicity breaking from quan- tum many-body scars, Nature Physics14, 745 (2018)

    2018

  88. [96]

    Moudgalya, S

    S. Moudgalya, S. Rachel, B. A. Bernevig, and N. Reg- nault, Exact excited states of nonintegrable models, Phys. Rev. B98, 235155 (2018)

    2018

  89. [97]

    Moudgalya, N

    S. Moudgalya, N. Regnault, and B. A. Bernevig, Entan- glement of exact excited states of affleck-kennedy-lieb- tasaki models: Exact results, many-body scars, and vi- olation of the strong eigenstate thermalization hypoth- esis, Phys. Rev. B98, 235156 (2018)

    2018

  90. [98]

    S. Choi, C. J. Turner, H. Pichler, W. W. Ho, A. A. Michailidis, Z. Papi´ c, M. Serbyn, M. D. Lukin, and D. A. Abanin, Emergent su(2) dynamics and perfect quantum many-body scars, Phys. Rev. Lett.122, 220603 (2019)

    2019

  91. [99]

    Serbyn, D

    M. Serbyn, D. A. Abanin, and Z. Papi´ c, Quantum many-body scars and weak breaking of ergodicity, Na- ture Physics17, 675 (2021)

    2021

  92. [100]

    Ge, Y.-R

    Z.-Y. Ge, Y.-R. Zhang, and F. Nori, Nonmesonic quan- tum many-body scars in a 1d lattice gauge theory, Phys. Rev. Lett.132, 230403 (2024)

    2024

  93. [101]

    D. V. Else, B. Bauer, and C. Nayak, Floquet time crys- tals, Phys. Rev. Lett.117, 090402 (2016)

    2016

  94. [102]

    N. Y. Yao, A. C. Potter, I.-D. Potirniche, and A. Vish- wanath, Discrete time crystals: Rigidity, criticality, and realizations, Phys. Rev. Lett.118, 030401 (2017)

    2017

  95. [103]

    Zhang, P

    J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter, A. Vishwanath, N. Y. Yao, and C. Monroe, Observation of a discrete time crystal, Nature543, 217 (2017)

    2017

  96. [104]

    Liu, S.-X

    S. Liu, S.-X. Zhang, C.-Y. Hsieh, S. Zhang, and H. Yao, Discrete time crystal enabled by stark many-body lo- calization, Phys. Rev. Lett.130, 120403 (2023). 8

    2023

  97. [105]

    P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity breaking arising from hilbert space fragmentation in dipole-conserving hamiltonians, Phys. Rev. X10, 011047 (2020)

    2020

  98. [106]

    Khemani, M

    V. Khemani, M. Hermele, and R. Nandkishore, Local- ization from hilbert space shattering: From theory to physical realizations, Phys. Rev. B101, 174204 (2020)

    2020

  99. [107]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, Hilbert space frag- mentation and commutant algebras, Phys. Rev. X12, 011050 (2022)

    2022

  100. [108]

    M. Will, R. Moessner, and F. Pollmann, Realization of hilbert space fragmentation and fracton dynamics in two dimensions, Phys. Rev. Lett.133, 196301 (2024)

    2024

  101. [109]

    B.-T. Chen, A. Prem, N. Regnault, and B. Lian, Quan- tum fragmentation in the extended quantum breakdown model, Phys. Rev. B110, 165109 (2024)

    2024

  102. [110]

    Zhou, T.-H

    Z. Zhou, T.-H. Yang, and B.-T. Chen, Quantum hilbert space fragmentation and entangled frozen states, arXiv:2604.05218 (2026)

    2026 arXiv

  103. [111]

    M. G. Genoni, M. G. A. Paris, and K. Banaszek, Quan- tifying the non-gaussian character of a quantum state by quantum relative entropy, Phys. Rev. A78, 060303(R) (2008)

    2008

  104. [112]

    Takagi and Q

    R. Takagi and Q. Zhuang, Convex resource theory of non-gaussianity, Phys. Rev. A97, 062337 (2018)

    2018

  105. [113]

    Albarelli, M

    F. Albarelli, M. G. Genoni, M. G. A. Paris, and A. Fer- raro, Resource theory of quantum non-gaussianity and wigner negativity, Phys. Rev. A98, 052350 (2018)

    2018

  106. [114]

    T. Haug, X. Turkeshi, and P. Sierant, Practi- cal tests and witnesses of fermionic non-gaussianity, arXiv:2605.26218 (2026)

    2026 arXiv

  107. [115]

    Paviglianiti, L

    A. Paviglianiti, L. Lumia, E. Tirrito, A. Silva, M. Col- lura, X. Turkeshi, and G. Lami, Emergence of generic entanglement structure in doped matchgate circuits, Phys. Rev. Lett.136, 020403 (2026)

    2026

  108. [116]

    Baumgratz, M

    T. Baumgratz, M. Cramer, and M. B. Plenio, Quanti- fying coherence, Phys. Rev. Lett.113, 140401 (2014)

    2014

  109. [117]

    Streltsov, G

    A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys.89, 041003 (2017)

    2017

  110. [118]

    Chitambar and G

    E. Chitambar and G. Gour, Critical examination of in- coherent operations and a physically consistent resource theory of quantum coherence, Phys. Rev. Lett.117, 030401 (2016)

    2016

  111. [119]

    Aditya, E

    S. Aditya, E. Tirrito, P. Sierant, and X. Turkeshi, Co- herence dynamics in quantum many-body systems with conservation laws, arXiv:2604.23192 (2026)

    2026 arXiv

  112. [120]

    D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett.71, 1291 (1993)

    1993

  113. [121]

    Lubkin, Entropy of an n-system from its correlation with a k-reservoir, Journal of Mathematical Physics19, 1028 (1978)

    E. Lubkin, Entropy of an n-system from its correlation with a k-reservoir, Journal of Mathematical Physics19, 1028 (1978)

    1978

  114. [122]

    Zyczkowski and H.-J

    K. Zyczkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, Journal of Physics A: Mathematical and General34, 7111 (2001)

    2001

  115. [123]

    Sommers and K

    H.-J. Sommers and K. ˙Zyczkowski, Statistical proper- ties of random density matrices, Journal of Physics A: Mathematical and General37, 8457 (2004)

    2004

  116. [124]

    A. A. Mele, Introduction to Haar Measure Tools in Quantum Information: A Beginner’s Tutorial, Quan- tum8, 1340 (2024)

    2024

  117. [125]

    Zhang, J

    S.-X. Zhang, J. Allcock, Z.-Q. Wan, S. Liu, J. Sun, H. Yu, X.-H. Yang, J. Qiu, Z. Ye, Y.-Q. Chen, C.-K. Lee, Y.-C. Zheng, S.-K. Jian, H. Yao, C.-Y. Hsieh, and S. Zhang, TensorCircuit: a Quantum Software Frame- work for the NISQ Era, Quantum7, 912 (2023)

    2023

  118. [126]

    Revealing Entanglement-Growth Mechanisms through the Magic Barrier

    S.-X. Zhang, Y.-Q. Chen, W. Li, J. Sun, W.-G. Ma, P.-L. Zheng, Y.-X. Huang, Q.-X. Wang, H. Yu, Z. Li, X. Huang, Z.-L. Li, Z.-Q. Wan, S. Liu, J. Qiu, J. Miao, Z. Song, Y. Yan, K. Tsuoka, P. Zhang, L. Wang, H. Fan, C.-Y. Hsieh, H. Yao, and T. Xiang, Tensorcircuit-ng: A universal...

    2026

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.