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arxiv: 2506.09904 · v3 · submitted 2025-06-11 · 🪐 quant-ph

Entanglement structure for finite system under dual-unitary dynamics

Gaurav Rudra Malik , Rohit Kumar Shukla , Sudhanva Joshi , S. Aravinda , Sunil Kumar Mishra This is my paper

Pith reviewed 2026-05-19 09:22 UTC · model grok-4.3

classification 🪐 quant-ph
keywords dual-unitary circuitsentanglement generationmultipartite entanglementquantum chaosinformation scramblingAME statesfinite quantum systems
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0 comments X p. Extension

The pith

Time-evolving pair-product initial states under dual-unitary circuits produces states with nearly maximal multipartite entanglement approaching AME bounds.

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

The paper examines how entanglement spreads in finite many-body systems modeled by dual-unitary circuits, a tractable framework for maximally chaotic quantum dynamics. It analyzes the influence of two-body operators on both bipartite and multipartite entanglement, while showing that local unitaries can produce different growth rates even when entangling power is fixed. Time-dependent lower bounds are derived that depend on the initial state and the gates' properties. A central result is that an initial state built from product pairs evolves to a configuration whose multipartite entanglement content nears the limits set by absolutely maximally entangled states.

Core claim

In dual-unitary circuits, time-evolving an initial state composed of pair products generates a state with nearly maximal multipartite entanglement content, approaching the bounds established by Absolutely Maximally Entangled (AME) states. The circuits allow independent control of the entangling power of constituent operators while preserving overall maximal chaos, and the paper establishes corresponding time-step-dependent lower bounds on entanglement measures.

What carries the argument

Dual-unitary gates whose entangling power can be varied independently while the overall dynamics remain maximally chaotic, combined with an initial state of pair products.

If this is right

  • Local unitaries paired with dual-unitary operators can produce differing entanglement growth rates even at fixed entangling power.
  • Time-step-dependent lower bounds on entanglement are set by the initial state and the operators' entangling power.
  • Bipartite and multipartite entanglement both increase under the dynamics, with the latter approaching AME limits.
  • The structure of the evolved state is shaped by the specific two-body operators chosen in the circuit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction may provide a controllable testbed for studying how information scrambles in larger chaotic spin systems.
  • Approaching AME bounds in finite systems could guide the design of short-depth circuits for tasks that require strong multipartite correlations.
  • Varying system size while keeping the same gate family would test whether the near-maximal entanglement persists or saturates as the chain length grows.

Load-bearing premise

Dual-unitary gates serve as an effective model for maximally chaotic dynamics in which entangling power can be tuned separately from the overall chaotic character.

What would settle it

Numerical computation of a multipartite entanglement measure on the evolved state for several system sizes and gate choices that falls well below the AME upper bound would falsify the central claim.

read the original abstract

The dynamics of quantum many-body systems in the chaotic regime are of particular interest due to the associated phenomena of information scrambling and entanglement generation within the system. While these systems are typically intractable using traditional numerical methods, an effective framework can be implemented based on dual-unitary circuits which have emerged as a minimal model for maximally chaotic dynamics. In this work, we investigate how individual two-body operators influence the global dynamics of circuits composed of dual-unitaries. We study their effect on entanglement generation while examining it from both bipartite and multipartite perspectives. Here we also highlight the significant role of local unitaries in the dynamics when paired with operators from the dual-unitary class, showing that systems with identical entangling power can exhibit a range of differing entanglement growth rates. Furthermore, we present calculations establishing time-step-dependent lower bounds, which depend on both the initial state and the entangling power of the constituent operators. Finally, we find that time-evolving an initial state composed of pair products generates a state with nearly maximal multipartite entanglement content, approaching the bounds established by Absolutely Maximally Entangled (AME) states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates entanglement generation in finite quantum many-body systems under dual-unitary circuit dynamics as a model for chaotic evolution. It examines the influence of individual two-body operators on global dynamics from both bipartite and multipartite perspectives, highlights the role of local unitaries in producing differing entanglement growth rates for operators with identical entangling power, derives time-step-dependent lower bounds on entanglement that depend on the initial state and the entangling power of the gates, and reports that time evolution of an initial state composed of pair products produces a state with nearly maximal multipartite entanglement content that approaches the bounds set by Absolutely Maximally Entangled (AME) states.

Significance. If the central results hold, the work provides a concrete, analytically tractable framework for studying information scrambling and multipartite entanglement production in chaotic quantum circuits. The explicit time-dependent lower bounds and the identification of pair-product initial states as generators of near-AME entanglement constitute useful, falsifiable contributions that could guide both theoretical understanding and experimental design of entanglement-maximizing protocols. The emphasis on the independent variation of entangling power while preserving maximal chaos is a strength of the modeling approach.

major comments (1)
  1. [§ on multipartite entanglement and AME comparison (near the end of the results)] § on multipartite entanglement and AME comparison (near the end of the results): the central claim that the evolved state 'approaches the bounds established by Absolutely Maximally Entangled (AME) states' is load-bearing. AME states require that the entanglement entropy saturates log(min(d_A,d_B)) for every bipartition. The manuscript must specify the precise multipartite diagnostic employed (e.g., average over partitions, a specific monotone, or a lower bound) and demonstrate that the bound is tight and uniform across all cuts rather than selected or averaged ones; otherwise the conclusion that the state approaches the full AME property for finite N does not follow.
minor comments (2)
  1. The abstract states that lower bounds 'depend on both the initial state and the entangling power' but does not indicate the explicit functional form or the range of entangling powers considered; a brief equation or table reference in the abstract would improve clarity.
  2. [Introduction or methods section] Notation for the entangling power of the two-body operators should be introduced with a clear definition on first use in the main text to avoid ambiguity when comparing different operators.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address the major comment below and will incorporate clarifications in a revised version.

read point-by-point responses

  1. Referee: the central claim that the evolved state 'approaches the bounds established by Absolutely Maximally Entangled (AME) states' is load-bearing. AME states require that the entanglement entropy saturates log(min(d_A,d_B)) for every bipartition. The manuscript must specify the precise multipartite diagnostic employed (e.g., average over partitions, a specific monotone, or a lower bound) and demonstrate that the bound is tight and uniform across all cuts rather than selected or averaged ones; otherwise the conclusion that the state approaches the full AME property for finite N does not follow.

    Authors: We thank the referee for this important observation. In the manuscript the multipartite diagnostic is the average bipartite entanglement entropy taken over all possible bipartitions (with the time-dependent lower bound derived from the initial pair-product state and the gate entangling power). This average is shown to approach the AME saturation value log(min(d_A,d_B)) at late times for the chosen initial states. We do not claim, nor does the data support, that every individual bipartition saturates the bound simultaneously for finite N; such uniform saturation is a stronger property typically realized only by specific AME constructions in certain dimensions. To remove any ambiguity we will revise the relevant section to (i) state explicitly that the diagnostic is the average entanglement entropy, (ii) emphasize that the approach is to the AME bound in the averaged sense together with the derived lower bounds, and (iii) add a brief remark distinguishing this from exact AME saturation across all cuts. These changes will be made in the next version. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from initial state and entangling power without self-referential reduction

full rationale

The paper computes time-step-dependent lower bounds on entanglement that are explicitly parameterized by the choice of initial state (pair-product form) and the tunable entangling power of the dual-unitary gates. These quantities are inputs to the calculation rather than outputs defined in terms of the final multipartite measure. The approach to AME bounds is reported as a numerical or analytic finding for that specific class of states, not as a definitional identity or a fitted parameter renamed as a prediction. No self-citation chain, uniqueness theorem, or ansatz smuggling is invoked to close the derivation; the dual-unitary framework is treated as an external minimal model for chaos. The derivation chain therefore remains independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on the dual-unitary framework as a minimal model for chaotic dynamics and on the definition of entangling power; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Dual-unitary circuits provide an effective minimal model for maximally chaotic dynamics in quantum many-body systems.
    Stated in the abstract as the basis for the framework.

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages · 1 internal anchor

  1. [1]

    Quantum en- tanglement,

    Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki, “Quantum en- tanglement,” Rev. Mod. Phys. 81, 865–942 (2009)

    work page 2009

  2. [2]

    Quantum information and computation,

    Charles H. Bennett and David P . DiVincenzo, “Quantum information and computation,” Nature (London) 404, 247–255 (2000)

    work page 2000

  3. [3]

    Demonstrat- ing the viability of universal quantum computation using teleportation and single-qubit operations,

    Daniel Gottesman and Isaac L. Chuang, “Demonstrat- ing the viability of universal quantum computation using teleportation and single-qubit operations,” Nature (Lon- don) 402, 390–393 (1999)

    work page 1999

  4. [4]

    Quantum computation and shor’s factoring algorithm,

    Artur Ekert and Richard Jozsa, “Quantum computation and shor’s factoring algorithm,” Rev. Mod. Phys.68, 733– 753 (1996)

    work page 1996

  5. [5]

    Quantum cryptography with entangled photons,

    Thomas Jennewein, Christoph Simon, Gregor Weihs, Harald Weinfurter, and Anton Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000)

    work page 2000

  6. [6]

    Colloquium: Area laws for the entanglement entropy,

    J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy,” Rev. Mod. Phys. 82, 277–306 (2010)

    work page 2010

  7. [7]

    A practical introduction to tensor net- works: Matrix product states and projected entangled pair states,

    Rom ´an Or ´us, “A practical introduction to tensor net- works: Matrix product states and projected entangled pair states,” Annals of Physics 349, 117–158 (2014)

    work page 2014

  8. [8]

    The density-matrix renormalization group in the age of matrix product states,

    Ulrich Schollw ¨ock, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011)

    work page 2011

  9. [9]

    Many-body lo- calization and thermalization in quantum statistical me- chanics,

    Rahul Nandkishore and David A. Huse, “Many-body lo- calization and thermalization in quantum statistical me- chanics,” Annual Review of Condensed Matter Physics 6, 15–38 (2015)

    work page 2015

  10. [10]

    Quantum many-body systems in thermal equilibrium,

    ´Alvaro M. Alhambra, “Quantum many-body systems in thermal equilibrium,” PRX Quantum 4, 040201 (2023)

    work page 2023

  11. [11]

    Measuring the scrambling of quan- tum information,

    Brian Swingle, Gregory Bentsen, Monika Schleier-Smith, and Patrick Hayden, “Measuring the scrambling of quan- tum information,” Phys. Rev. A 94, 040302 (2016)

    work page 2016

  12. [12]

    Ergodic properties of a generic nonin- tegrable quantum many-body system in the thermody- namic limit,

    Toma ˇz Prosen, “Ergodic properties of a generic nonin- tegrable quantum many-body system in the thermody- namic limit,” Phys. Rev. E 60, 3949–3968 (1999)

    work page 1999

  13. [13]

    Chaos and complexity of quantum mo- tion,

    Toma ˇz Prosen, “Chaos and complexity of quantum mo- tion,” Journal of Physics A: Mathematical and Theoreti- cal 40, 7881 (2007)

    work page 2007

  14. [14]

    Colloquium: Many-body localization, thermalization, and entanglement,

    Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn, “Colloquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. 91, 021001 (2019)

    work page 2019

  15. [15]

    Many-body localization in a disordered quantum ising chain,

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

    work page 2014

  16. [16]

    Scrambling of quan- tum information in quantum many-body systems,

    Eiki Iyoda and Takahiro Sagawa, “Scrambling of quan- tum information in quantum many-body systems,” Phys. Rev. A 97, 042330 (2018)

    work page 2018

  17. [17]

    Scrambling in ising spin systems with constant and periodic transverse magnetic fields,

    Rohit Kumar Shukla, “Scrambling in ising spin systems with constant and periodic transverse magnetic fields,” arXiv preprint arXiv:2310.14620 (2023)

    work page arXiv 2023

  18. [18]

    Out-of-time- ordered correlators in a quantum ising chain,

    Cheng-Ju Lin and Olexei I. Motrunich, “Out-of-time- ordered correlators in a quantum ising chain,” Phys. Rev. B 97, 144304 (2018). 18

    work page 2018

  19. [19]

    Out-of-time-order correlators of nonlocal block-spin and random observables in inte- grable and nonintegrable spin chains,

    Rohit Kumar Shukla, Arul Lakshminarayan, and Sunil Kumar Mishra, “Out-of-time-order correlators of nonlocal block-spin and random observables in inte- grable and nonintegrable spin chains,” Phys. Rev. B 105, 224307 (2022)

    work page 2022

  20. [20]

    Charac- teristic, dynamic, and near-saturation regions of out-of- time-order correlation in floquet ising models,

    Rohit Kumar Shukla and Sunil Kumar Mishra, “Charac- teristic, dynamic, and near-saturation regions of out-of- time-order correlation in floquet ising models,” Physical Review A 106, 022403 (2022)

    work page 2022

  21. [21]

    A bound on chaos,

    Juan Maldacena, Stephen H. Shenker, and Douglas Stanford, “A bound on chaos,” Journal of High Energy Physics 2016, 106 (2016)

    work page 2016

  22. [22]

    Oper- ator spreading in random unitary circuits,

    Adam Nahum, Sagar Vijay, and Jeongwan Haah, “Oper- ator spreading in random unitary circuits,” Phys. Rev. X 8, 021014 (2018)

    work page 2018

  23. [23]

    Quantum entanglement growth under random unitary dynamics,

    Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah, “Quantum entanglement growth under random unitary dynamics,” Phys. Rev. X7, 031016 (2017)

    work page 2017

  24. [24]

    Operator spreading and the emergence of dissi- pative hydrodynamics under unitary evolution with con- servation laws,

    Vedika Khemani, Ashvin Vishwanath, and David A. Huse, “Operator spreading and the emergence of dissi- pative hydrodynamics under unitary evolution with con- servation laws,” Phys. Rev. X 8, 031057 (2018)

    work page 2018

  25. [25]

    Operator hydrodynamics, otocs, and entanglement growth in systems without con- servation laws,

    C. W. von Keyserlingk, Tibor Rakovszky, Frank Poll- mann, and S. L. Sondhi, “Operator hydrodynamics, otocs, and entanglement growth in systems without con- servation laws,” Phys. Rev. X 8, 021013 (2018)

    work page 2018

  26. [26]

    Testing statistical bounds on entanglement using quan- tum chaos,

    Jayendra N. Bandyopadhyay and Arul Lakshminarayan, “Testing statistical bounds on entanglement using quan- tum chaos,” Phys. Rev. Lett. 89, 060402 (2002)

    work page 2002

  27. [27]

    Signatures of chaos in the entanglement of two coupled quantum kicked tops,

    Paul A. Miller and Sarben Sarkar, “Signatures of chaos in the entanglement of two coupled quantum kicked tops,” Phys. Rev. E 60, 1542–1550 (1999)

    work page 1999

  28. [28]

    Entanglement of quantum evolutions,

    Paolo Zanardi, “Entanglement of quantum evolutions,” Phys. Rev. A 63, 040304 (2001)

    work page 2001

  29. [29]

    Entan- gling power of quantum evolutions,

    Paolo Zanardi, Christof Zalka, and Lara Faoro, “Entan- gling power of quantum evolutions,” Phys. Rev. A 62, 030301 (2000)

    work page 2000

  30. [30]

    From dual-unitary to quantum bernoulli cir- cuits: Role of the entangling power in constructing a quantum ergodic hierarchy,

    S. Aravinda, Suhail Ahmad Rather, and Arul Lakshmi- narayan, “From dual-unitary to quantum bernoulli cir- cuits: Role of the entangling power in constructing a quantum ergodic hierarchy,” Phys. Rev. Res. 3, 043034 (2021)

    work page 2021

  31. [31]

    Information scrambling over bipartitions: Equilibration, entropy production, and typicality,

    Georgios Styliaris, Namit Anand, and Paolo Zanardi, “Information scrambling over bipartitions: Equilibration, entropy production, and typicality,” Phys. Rev. Lett. 126, 030601 (2021)

    work page 2021

  32. [32]

    Exactly solvable many-body dynamics from space-time duality

    Toma ˇz Prosen Bruno Bertini, Pieter W. Claeys, “Exactly solvable many-body dynamics from space-time duality,” (2025), arXiv:2505.11489

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  33. [33]

    Exact correlation functions for dual-unitary lattice models in 1 + 1 dimensions,

    Bruno Bertini, Pavel Kos, and Toma ˇz Prosen, “Exact correlation functions for dual-unitary lattice models in 1 + 1 dimensions,” Phys. Rev. Lett. 123, 210601 (2019)

    work page 2019

  34. [34]

    Maximum ve- locity quantum circuits,

    Pieter W. Claeys and Austen Lamacraft, “Maximum ve- locity quantum circuits,” Phys. Rev. Res. 2, 033032 (2020)

    work page 2020

  35. [35]

    Entanglement membrane in exactly solvable lat- tice models,

    Michael A. Rampp, Suhail A. Rather, and Pieter W. Claeys, “Entanglement membrane in exactly solvable lat- tice models,” Phys. Rev. Res. 6, 033271 (2024)

    work page 2024

  36. [36]

    Exact dynamics in dual-unitary quan- tum circuits,

    Lorenzo Piroli, Bruno Bertini, J. Ignacio Cirac, and Toma ˇz Prosen, “Exact dynamics in dual-unitary quan- tum circuits,” Phys. Rev. B 101, 094304 (2020)

    work page 2020

  37. [37]

    Growth of entan- glement of generic states under dual-unitary dynamics,

    Alessandro Foligno and Bruno Bertini, “Growth of entan- glement of generic states under dual-unitary dynamics,” Phys. Rev. B 107, 174311 (2023)

    work page 2023

  38. [38]

    Computational power of one- and two-dimensional dual-unitary quantum circuits,

    Ryotaro Suzuki, Kosuke Mitarai, and Keisuke Fujii, “Computational power of one- and two-dimensional dual-unitary quantum circuits,” Quantum 6, 631 (2022)

    work page 2022

  39. [39]

    Quantum many-body scars in dual-unitary circuits,

    Leonard Logari ´c, Shane Dooley, Silvia Pappalardi, and John Goold, “Quantum many-body scars in dual-unitary circuits,” Phys. Rev. Lett. 132, 010401 (2024)

    work page 2024

  40. [40]

    Impact of local dynamics on entangling power,

    Bhargavi Jonnadula, Prabha Mandayam, Karol ˙Zyczkowski, and Arul Lakshminarayan, “Impact of local dynamics on entangling power,” Phys. Rev. A 95, 040302 (2017)

    work page 2017

  41. [41]

    Entangling power of time-evolution operators in integrable and non- integrable many-body systems,

    Rajarshi Pal and Arul Lakshminarayan, “Entangling power of time-evolution operators in integrable and non- integrable many-body systems,” Phys. Rev. B 98, 174304 (2018)

    work page 2018

  42. [42]

    Multipar- tite entanglement in a one-dimensional time-dependent ising model,

    Arul Lakshminarayan and V . Subrahmanyam, “Multipar- tite entanglement in a one-dimensional time-dependent ising model,” Phys. Rev. A 71, 062334 (2005)

    work page 2005

  43. [43]

    Entanglement as a signature of quantum chaos,

    Xiaoguang Wang, Shohini Ghose, Barry C. Sanders, and Bambi Hu, “Entanglement as a signature of quantum chaos,” Phys. Rev. E 70, 016217 (2004)

    work page 2004

  44. [44]

    Ingemar Bengtsson and Karol Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, 2006)

    work page 2006

  45. [45]

    Entangle- ment measures of bipartite quantum gates and their thermalization under arbitrary interaction strength,

    Bhargavi Jonnadula, Prabha Mandayam, Karol ˙Zyczkowski, and Arul Lakshminarayan, “Entangle- ment measures of bipartite quantum gates and their thermalization under arbitrary interaction strength,” Phys. Rev. Res. 2, 043126 (2020)

    work page 2020

  46. [46]

    Absolute maximal entan- glement and quantum secret sharing,

    Wolfram Helwig, Wei Cui, Jos ´e Ignacio Latorre, Arnau Riera, and Hoi-Kwong Lo, “Absolute maximal entan- glement and quantum secret sharing,” Phys. Rev. A 86, 052335 (2012)

    work page 2012

  47. [47]

    Thirty-six entangled officers of euler: Quantum solution to a classically im- possible problem,

    Suhail Ahmad Rather, Adam Burchardt, Wojciech Bruzda, Grzegorz Rajchel-Mieldzio ´c, Arul Lakshmi- narayan, and Karol ˙Zyczkowski, “Thirty-six entangled officers of euler: Quantum solution to a classically im- possible problem,” Phys. Rev. Lett. 128, 080507 (2022)

    work page 2022

  48. [48]

    How entangled can two couples get?

    A. Higuchi and A. Sudbery, “How entangled can two couples get?” Physics Letters A 273, 213–217 (2000)

    work page 2000

  49. [49]

    Matrix realignment and partial-transpose approach to entangling power of quantum evolutions,

    Zhihao Ma and Xiaoguang Wang, “Matrix realignment and partial-transpose approach to entangling power of quantum evolutions,” Phys. Rev. A 75, 014304 (2007)

    work page 2007

  50. [50]

    Quantum entan- glement of unitary operators on bipartite systems,

    Xiaoguang Wang and Paolo Zanardi, “Quantum entan- glement of unitary operators on bipartite systems,” Phys. Rev. A 66, 044303 (2002)

    work page 2002

  51. [51]

    Introduction to Haar Measure Tools in Quantum Information: A Beginner’s Tutorial,

    Antonio Anna Mele, “Introduction to Haar Measure Tools in Quantum Information: A Beginner’s Tutorial,” Quantum 8, 1340 (2024)

    work page 2024

  52. [52]

    Optimal creation of entangle- ment using a two-qubit gate,

    B. Kraus and J. I. Cirac, “Optimal creation of entangle- ment using a two-qubit gate,” Phys. Rev. A 63, 062309 (2001)

    work page 2001

  53. [53]

    Entangling power and local invariants of two-qubit gates,

    S. Balakrishnan and R. Sankaranarayanan, “Entangling power and local invariants of two-qubit gates,” Phys. Rev. A 82, 034301 (2010)

    work page 2010

  54. [54]

    Creating ensembles of dual unitary and max- imally entangling quantum evolutions,

    Suhail Ahmad Rather, S. Aravinda, and Arul Lakshmi- narayan, “Creating ensembles of dual unitary and max- imally entangling quantum evolutions,” Phys. Rev. Lett. 125, 070501 (2020). 19

    work page 2020

  55. [55]

    Solu- tion of a minimal model for many-body quantum chaos,

    Amos Chan, Andrea De Luca, and J. T. Chalker, “Solu- tion of a minimal model for many-body quantum chaos,” Phys. Rev. X 8, 041019 (2018)

    work page 2018

  56. [56]

    Entangle- ment spreading in a minimal model of maximal many- body quantum chaos,

    Bruno Bertini, Pavel Kos, and Toma ˇz Prosen, “Entangle- ment spreading in a minimal model of maximal many- body quantum chaos,” Phys. Rev. X 9, 021033 (2019)

    work page 2019

  57. [57]

    Unitary circuits of finite depth and infinite width from quantum channels,

    Sarang Gopalakrishnan and Austen Lamacraft, “Unitary circuits of finite depth and infinite width from quantum channels,” Phys. Rev. B 100, 064309 (2019)

    work page 2019

  58. [58]

    Multipartite entanglement, quantum-error- correcting codes, and entangling power of quantum evo- lutions,

    A. J. Scott, “Multipartite entanglement, quantum-error- correcting codes, and entangling power of quantum evo- lutions,” Phys. Rev. A 69, 052330 (2004)

    work page 2004

  59. [59]

    Multipartite entanglement measure : Genuine to absolutely maximally entangled,

    Rahul V . and Aravinda S., “Multipartite entanglement measure : Genuine to absolutely maximally entangled,” (2024), arXiv:2410.15864

    work page arXiv 2024

  60. [60]

    Quantum circuits for high- dimensional absolutely maximally entangled states,

    Berta Casas et.al., “Quantum circuits for high- dimensional absolutely maximally entangled states,” (2025), arXiv:2504.05394

    work page arXiv 2025

  61. [61]

    Operator space entangling power of quantum dy- namics and local operator entanglement growth in dual- unitary circuits,

    Faidon Andreadakis, Emanuel Dallas, and Paolo Za- nardi, “Operator space entangling power of quantum dy- namics and local operator entanglement growth in dual- unitary circuits,” Phys. Rev. A 110, 052416 (2024)

    work page 2024

  62. [62]

    Quantum chaos challenges many-body localiza- tion,

    Jan Suntajs, Janez Bonca, Tomaz Prosen, and Lev Vid- mar, “Quantum chaos challenges many-body localiza- tion,” Phys. Rev. E 102, 062144 (2020). Appendix A: Creating Examples of Dual Unitaries

    work page 2020

  63. [63]

    Qubit Case In the qubit case it is straight-forward to generate U ∈ DU (2), following the Cartan decomposition of two-qubit unitary gates [ 52]. For a two qubit operator A, we have: A = eiϕ(u+ ⊗ u−)V[J1, J2, J3](v+ ⊗ v−) (A1) where ϕ, J1, J2, J3 ∈ R and the local unitary operators u± and v± belong to the general SU( 2) group. Here V(J1, J2, J3) is given a...

    work page

  64. [64]

    However, there exist several ways to get operators U ∈ DU (3), some of which also ∈ U 2(3)

    Qutrit Case While the qubit case is straightforward in terms of its relation with a physical many-body hamiltonian, a sim- ilar approach is not know for the case q = 3. However, there exist several ways to get operators U ∈ DU (3), some of which also ∈ U 2(3). A common method relies on using permutations, which generate several dual-unitaries with differe...

    work page

  65. [65]

    R defines a linear step where R(U) = UR2

    work page

  66. [66]

    The operator V is found by the polar decomposi- tion of UR2 = VP, where V is the required uni- tary and P is a positive semi-definite Hermitian matrix

    P D(U) is the non-linear step, where P D(U) = V, which is the unitary closest to the operator UR2. The operator V is found by the polar decomposi- tion of UR2 = VP, where V is the required uni- tary and P is a positive semi-definite Hermitian matrix. In our case for generating qutrit dual unitaries we start off with an ensemble of a 1000 matrices sampled ...

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  67. [67]

    To begin, let us consider a finite circuit of N = 10 pair products (i.e

    Setting In this section we will describe the manner in which entanglement builds up for a finite system size during time evolution, following the graphical approach. To begin, let us consider a finite circuit of N = 10 pair products (i.e. L = 20) forming the initial state, which is then evolved for a single step. We then describe the density matrix ρA(t =...

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  68. [68]

    A particular example of the C′ x operator for x = 4, involving a specific U′ is given as follows

    Entanglement Velocity Bounds Calculating the ensemble average of Renyi entropy implies averaging over the scalar quantity having the form −2 ln tr h (C′ xC′† x)2 i for all members of the ensemble. A particular example of the C′ x operator for x = 4, involving a specific U′ is given as follows. Note that for each C′, the initial state is kept fixed. C′ x=4...

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