pith. sign in

arxiv: 2607.01351 · v1 · pith:CO45SCIEnew · submitted 2026-07-01 · ✦ hep-th

Wigner negativity in Krylov space and emergent semiclassicality

Vijay Balasubramanian , Pawel Caputa , Onkar Parrikar , Vivek Singh This is my paper

Pith reviewed 2026-07-03 19:24 UTC · model grok-4.3

classification ✦ hep-th
keywords Krylov basisWigner negativitysemiclassicalitylarge-N limit2d CFTrandom matrix theorySYK modelquantum dynamics
0
0 comments X

The pith

The Krylov basis makes dynamics in large-N many-body systems appear semiclassical, with Wigner negativity staying constant or growing only as the square root of time.

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

The paper proposes that the Krylov basis supplies a semiclassical representation of time evolution in general large-N complex many-body systems. The authors test this idea by measuring the growth of Wigner negativity, a quantity that tracks how difficult classical simulation becomes, during evolution within the Krylov basis. In 2d CFTs prepared in the vacuum or a thermofield double state and excited by a primary, negativity remains an O(1) constant with no late-time growth. In one-cut random matrix models with a maximally entangled initial state, negativity grows as t to the power 1/2 at intermediate times yet stays independent of Hilbert-space dimension. The double-scaled SYK model shows an early constant phase followed by the same slow t to the power 1/2 growth, indicating that semiclassical behavior emerges across these examples.

Core claim

We propose that the Krylov basis gives a semiclassical representation of dynamics in general large-N, complex, many-body systems. As a probe of this semiclassicality, we study the growth of Wigner negativity under time evolution in Krylov space in several solvable models. In 2d CFTs, negativity remains an O(1) constant and does not grow at late times. In general one-cut matrix models, negativity grows as t^{1/2} at large O(1) times but does not scale with the Hilbert space dimension. In the double-scaled SYK model, an approximately classical phase at early times is followed by a semiclassical phase with t^{1/2} growth at late times. In all cases, negativity either remains constant or grows s

What carries the argument

The Krylov basis, generated by repeated application of the Hamiltonian starting from an initial state, used here as the representation in which Wigner negativity is computed to test for semiclassical dynamics.

If this is right

  • In 2d CFTs with vacuum or thermofield-double initial states excited by primaries, Wigner negativity stays O(1) at all late times.
  • In one-cut random matrix models, negativity grows as t^{1/2} yet remains independent of system size.
  • Double-scaled SYK exhibits a constant-negativity phase at early times followed by t^{1/2} growth.
  • Across all examined cases, the slow or absent growth of negativity indicates that Krylov-space dynamics remain approximately classical.

Where Pith is reading between the lines

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

  • If the pattern holds beyond solvable models, classical simulation methods could be applied directly in Krylov space to approximate many-body evolution at large N.
  • The result suggests checking whether other operator bases or subspaces also suppress Wigner negativity in the same systems.
  • It would be useful to test the same negativity diagnostic in higher-dimensional field theories or lattice models where exact solutions are unavailable.

Load-bearing premise

The specific solvable models examined are representative of generic large-N many-body systems, and Wigner negativity serves as a reliable quantitative proxy for the difficulty of classical simulation.

What would settle it

A calculation in one of the models showing Wigner negativity that grows linearly with time or scales with Hilbert-space dimension in the Krylov basis would falsify the claim of emergent semiclassicality.

read the original abstract

We propose that the Krylov basis gives a semiclassical representation of dynamics in general large-$N$, complex, many-body systems. As a probe of this semiclassicality, we study the growth of Wigner negativity -- a measure of the complexity of classical simulation -- under time evolution in Krylov space in several solvable models. We begin with 2d CFTs, initially in either the vacuum or the thermofield double state on a line excited by a primary operator. In both cases, Wigner negativity remains an $O(1)$ constant and does not grow at late times, indicating approximately classical dynamics in the Krylov basis. We then study random matrix theory with the maximally entangled state between two copies as the initial state. For general one-cut matrix models, we argue that Wigner negativity in the Krylov basis grows as $t^{1/2}$ at large $O(1)$ times but does not scale with the Hilbert space dimension, thus indicating semiclassical dynamics in Krylov space. Finally, in the double-scaled SYK model, we find an approximately classical phase (constant negativity) at early times and a semiclassical phase ($t^{1/2}$ growth) at late times. In all these examples, Wigner negativity either remains constant or grows slowly, demonstrating emergent semiclassicality of dynamics in Krylov space.

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

2 major / 1 minor

Summary. The manuscript proposes that the Krylov basis provides a semiclassical representation of dynamics in general large-N, complex, many-body systems. This is probed by studying the growth of Wigner negativity under time evolution in Krylov space across three solvable models: 2d CFTs (vacuum or TFD + primary, where negativity remains O(1)), one-cut random matrix models (maximally entangled initial state, where negativity grows as t^{1/2} independent of Hilbert-space dimension), and the double-scaled SYK model (early-time constant negativity followed by late-time t^{1/2} growth).

Significance. If the central claim holds, the result would indicate emergent semiclassicality in Krylov space for large-N systems, with Wigner negativity serving as a quantitative proxy for the difficulty of classical simulation. The explicit calculations in the three solvable models constitute a concrete strength, yielding specific scaling predictions (O(1) or t^{1/2}) that are falsifiable within those models.

major comments (2)
  1. [§1] §1 (Introduction): The central claim that the Krylov basis yields semiclassical dynamics in 'general large-N, complex, many-body systems' extrapolates from explicit calculations in three models possessing special structure (conformal symmetry, random-matrix universality, all-to-all maximal chaos) without a universality argument or additional generic non-integrable, non-random example to establish that negativity growth remains sub-linear and dimension-independent outside this class.
  2. [§3] §3 (Random matrix models): The argument that Wigner negativity grows as t^{1/2} at large O(1) times but does not scale with Hilbert space dimension for general one-cut matrix models is tied to the maximally entangled initial state; it is unclear whether this independence from dimension follows from the model properties alone or requires additional assumptions that limit generality.
minor comments (1)
  1. Notation for the Krylov basis and Wigner negativity could be introduced with a brief self-contained definition in the main text rather than relying solely on references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our calculations and for the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§1] §1 (Introduction): The central claim that the Krylov basis yields semiclassical dynamics in 'general large-N, complex, many-body systems' extrapolates from explicit calculations in three models possessing special structure (conformal symmetry, random-matrix universality, all-to-all maximal chaos) without a universality argument or additional generic non-integrable, non-random example to establish that negativity growth remains sub-linear and dimension-independent outside this class.

    Authors: The manuscript presents the statement as a proposal ('We propose that...') motivated by the explicit results across three distinct classes of large-N models. We do not provide a universality argument or additional examples. To address the concern, we will revise the introduction to frame the claim more explicitly as a conjecture supported by the studied models rather than implying a general result. revision: partial

  2. Referee: [§3] §3 (Random matrix models): The argument that Wigner negativity grows as t^{1/2} at large O(1) times but does not scale with Hilbert space dimension for general one-cut matrix models is tied to the maximally entangled initial state; it is unclear whether this independence from dimension follows from the model properties alone or requires additional assumptions that limit generality.

    Authors: Section 3 explicitly uses the maximally entangled initial state between two copies. The t^{1/2} growth without dimension scaling is derived for one-cut models under this initial condition. We will revise the text to clarify the dependence on this choice of initial state and the associated assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on explicit calculations in solvable models

full rationale

The paper's proposal that the Krylov basis yields semiclassical dynamics for general large-N systems is supported by direct computations of Wigner negativity in three specific models: 2d CFTs (vacuum or TFD + primary, negativity O(1)), one-cut random matrices (maximally entangled state, ~t^{1/2} growth independent of dimension), and double-scaled SYK (early constant, late t^{1/2}). These are presented as evidence rather than derived from fitted parameters renamed as predictions or from self-citations. No self-definitional steps, load-bearing self-citations, or ansatze imported via citation appear in the derivation chain. The extrapolation to generic systems is an interpretive claim, not a reduction by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central proposal rests on the domain assumption that Krylov-space dynamics are representative of generic large-N systems and that Wigner negativity faithfully tracks classical simulability; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Krylov basis is obtained by repeated application of the Hamiltonian starting from a reference state
    Standard construction in Krylov-complexity literature invoked implicitly throughout the abstract.
  • domain assumption Wigner negativity quantifies the complexity of classical simulation
    Used as the central diagnostic without further justification in the abstract.

pith-pipeline@v0.9.1-grok · 5782 in / 1347 out tokens · 25576 ms · 2026-07-03T19:24:24.923399+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

84 extracted references · 52 canonical work pages · 18 internal anchors

  1. [1]

    Viswanath and G

    V. Viswanath and G. M¨ uller,The recursion method: application to many-body dynamics, Springer (1994)

    1994

  2. [2]

    Quantum Dynamics in Krylov Space: Methods and Applications

    P. Nandy, A.S. Matsoukas-Roubeas, P. Mart´ ınez-Azcona, A. Dymarsky and A. del Campo, Quantum dynamics in Krylov space: Methods and applications,Phys. Rept.1125-1128 (2025) 1 [2405.09628]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  3. [3]

    Krylov Complexity

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir and J. Sonner,Krylov Complexity,2507.06286

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    Baiguera, V

    S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M.P. Heller et al., Quantum complexity in gravity, quantum field theory, and quantum information science, 2503.10753

    work page arXiv

  5. [5]

    Balasubramanian, P

    V. Balasubramanian, P. Caputa, J.M. Magan and Q. Wu,Quantum chaos and the complexity of spread of states,Phys. Rev. D106(2022) 046007 [2202.06957]

    work page arXiv 2022

  6. [6]

    R. Basu, A. Ganguly, S. Nath and O. Parrikar,Complexity growth and the Krylov-Wigner function,Journal of High Energy Physics2024(2024) 264 [2402.13694]. – 39 –

    work page arXiv 2024

  7. [7]

    Veitch, S.A.H

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

    2014

  8. [8]

    Paeckel, T

    S. Paeckel, T. K¨ ohler, A. Swoboda, S.R. Manmana, U. Schollw¨ ock and C. Hubig, Time-evolution methods for matrix-product states,Annals Phys.411(2019) 167998 [1901.05824]

    work page arXiv 2019

  9. [9]

    R. Basu, P. Chowdhury, A. Ganguly, S. Nath, O. Parrikar and S. Paul,Wigner negativity, random matrices and gravity,2506.02110

    work page arXiv

  10. [10]

    Balasubramanian, J.M

    V. Balasubramanian, J.M. Magan and Q. Wu,Tridiagonalizing random matrices,Phys. Rev. D107(2023) 126001 [2208.08452]

    work page arXiv 2023

  11. [11]

    A. Kar, L. Lamprou, M. Rozali and J. Sully,Random matrix theory for complexity growth and black hole interiors,JHEP01(2022) 016 [2106.02046]

    work page arXiv 2022

  12. [12]

    Kuijlaars, K.T.-R

    A.B.J. Kuijlaars, K.T.-R. McLaughlin, W.V. Assche and M. Vanlessen,The riemann-hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1], 2003

    2003

  13. [13]

    Complexity and Shock Wave Geometries

    D. Stanford and L. Susskind,Complexity and Shock Wave Geometries,Phys. Rev. D90 (2014) 126007 [1406.2678]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  14. [14]

    Jefferson and R.C

    R.A. Jefferson and R.C. Myers,Circuit complexity in quantum field theory,Journal of High Energy Physics2017(2017)

    2017

  15. [15]

    Chapman, M.P

    S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski,Toward a definition of complexity for quantum field theory states,Physical Review Letters120(2018)

    2018

  16. [16]

    Three Lectures on Complexity and Black Holes

    L. Susskind,Three Lectures on Complexity and Black Holes, SpringerBriefs in Physics, Springer, 10, 2018, DOI [1810.11563]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  17. [17]

    Brown and L

    A.R. Brown and L. Susskind,Second law of quantum complexity,Physical Review D97 (2018)

    2018

  18. [18]

    Balasubramanian, M

    V. Balasubramanian, M. Decross, A. Kar and O. Parrikar,Quantum Complexity of Time Evolution with Chaotic Hamiltonians,JHEP01(2020) 134 [1905.05765]

    work page arXiv 2020

  19. [19]

    Balasubramanian, M

    V. Balasubramanian, M. DeCross, A. Kar, Y.C. Li and O. Parrikar,Complexity growth in integrable and chaotic models,JHEP07(2021) 011 [2101.02209]

    work page arXiv 2021

  20. [20]

    Towards a full solution of the large N double-scaled SYK model

    M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torrents,Towards a full solution of the large N double-scaled SYK model,JHEP03(2019) 079 [1811.02584]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  21. [21]

    Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction

    M. Berkooz, P. Narayan and J. Simon,Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction,JHEP08(2018) 192 [1806.04380]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  22. [22]

    Lin,The bulk Hilbert space of double scaled SYK,JHEP11(2022) 060 [2208.07032]

    H.W. Lin,The bulk Hilbert space of double scaled SYK,JHEP11(2022) 060 [2208.07032]

    work page arXiv 2022

  23. [23]

    Rabinovici, A

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir and J. Sonner,A bulk manifestation of Krylov complexity,JHEP08(2023) 213 [2305.04355]

    work page arXiv 2023

  24. [24]

    Balasubramanian, J.M

    V. Balasubramanian, J.M. Magan, P. Nandi and Q. Wu,Spread complexity and the saturation of wormhole size,Phys. Rev. D113(2026) 046004 [2412.02038]

    work page arXiv 2026

  25. [25]

    Balasubramanian, P

    V. Balasubramanian, P. Caputa and J. Sim´ on,Variations on a Theme of Krylov, 2511.03775

    work page arXiv

  26. [26]

    Gross,Hudson’s theorem for finite-dimensional quantum systems,Journal of Mathematical Physics47(2006)

    D. Gross,Hudson’s theorem for finite-dimensional quantum systems,Journal of Mathematical Physics47(2006) . – 40 –

    2006

  27. [27]

    Wootters,A wigner-function formulation of finite-state quantum mechanics,Annals of Physics176(1987) 1

    W.K. Wootters,A wigner-function formulation of finite-state quantum mechanics,Annals of Physics176(1987) 1

    1987

  28. [28]

    Leonhardt,Quantum-state tomography and discrete wigner function,Phys

    U. Leonhardt,Quantum-state tomography and discrete wigner function,Phys. Rev. Lett.74 (1995) 4101

    1995

  29. [29]

    Heiss and S

    S. Heiss and S. Weigert,Discrete moyal-type representations for a spin,Phys. Rev. A63 (2000) 012105

    2000

  30. [30]

    Miquel, J.P

    C. Miquel, J.P. Paz and M. Saraceno,Quantum computers in phase space,Phys. Rev. A65 (2002) 062309

    2002

  31. [31]

    Gibbons, M.J

    K.S. Gibbons, M.J. Hoffman and W.K. Wootters,Discrete phase space based on finite fields, Physical Review A70(2004)

    2004

  32. [32]

    Cormick, E.F

    C. Cormick, E.F. Galv˜ ao, D. Gottesman, J.P. Paz and A.O. Pittenger,Classicality in discrete wigner functions,Phys. Rev. A73(2006) 012301

    2006

  33. [33]

    Wigner,On the quantum correction for thermodynamic equilibrium,Phys

    E. Wigner,On the quantum correction for thermodynamic equilibrium,Phys. Rev.40(1932) 749

    1932

  34. [34]

    Gottesman,The heisenberg representation of quantum computers, 1998

    D. Gottesman,The heisenberg representation of quantum computers, 1998

    1998

  35. [35]

    Improved Simulation of Stabilizer Circuits

    S. Aaronson and D. Gottesman,Improved simulation of stabilizer circuits,Phys. Rev. A70 (2004) 052328 [quant-ph/0406196]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  36. [36]

    Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic

    E. Hostens, J. Dehaene and B.D. Moor,Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic,Phys. Rev. A71(2005) 042315 [quant-ph/0408190]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  37. [37]

    Bravyi and A

    S. Bravyi and A. Kitaev,Universal quantum computation with ideal clifford gates and noisy ancillas,Physical Review A71(2005)

    2005

  38. [38]

    White, C

    C.D. White, C. Cao and B. Swingle,Conformal field theories are magical,Phys. Rev. B103 (2021) 075145 [2007.01303]

    work page arXiv 2021

  39. [39]

    Cao,Non-trivial area operators require non-local magic,JHEP11(2024) 105 [2306.14996]

    C. Cao,Non-trivial area operators require non-local magic,JHEP11(2024) 105 [2306.14996]

    work page arXiv 2024

  40. [40]

    C. Cao, G. Cheng, A. Hamma, L. Leone, W. Munizzi and S.F.E. Oliviero,Gravitational back-reaction is magical,2403.07056

    work page arXiv

  41. [41]

    R. Basu, O. Parrikar, S. Paul and H. Rajgadia,On the stabilizer complexity of Hawking radiation,2510.18967

    work page arXiv

  42. [42]

    Malvimat, M

    V. Malvimat, M. Sarkis, Y. Suk and J. Yoon,Multipartite Non-local Magic and SYK Model, 2601.03076

    work page arXiv

  43. [43]

    Magic and Wormholes in the Sachdev-Ye-Kitaev Model

    V. Bettaque and B. Swingle,Magic and Wormholes in the Sachdev-Ye-Kitaev Model, 2602.12339

    work page internal anchor Pith review Pith/arXiv arXiv

  44. [44]

    Efficient classical simulation of slightly entangled quantum computations

    G. Vidal,Efficient Classical Simulation of Slightly Entangled Quantum Computations,Phys. Rev. Lett.91(2003) 147902 [quant-ph/0301063]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  45. [45]

    Yaffe,Large n Limits as Classical Mechanics,Rev

    L.G. Yaffe,Large n Limits as Classical Mechanics,Rev. Mod. Phys.54(1982) 407

    1982

  46. [46]

    Jevicki and B

    A. Jevicki and B. Sakita,The quantum collective field method and its application to the planar limit,Nuclear Physics B165(1980) 511

    1980

  47. [47]

    Das and A

    S.R. Das and A. Jevicki,String Field Theory and Physical Interpretation ofD= 1Strings, Mod. Phys. Lett. A5(1990) 1639. – 41 –

    1990

  48. [48]

    M¨ uck and Y

    W. M¨ uck and Y. Yang,Krylov complexity and orthogonal polynomials,Nucl. Phys. B984 (2022) 115948 [2205.12815]

    work page arXiv 2022

  49. [49]

    Balasubramanian, R.N

    V. Balasubramanian, R.N. Das, J. Erdmenger, J. Karl and H. Verlinde,Complexity and the Hilbert space dimension of 3D gravity,2602.02645

    work page arXiv

  50. [50]

    Parker, X

    D.E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman,A Universal Operator Growth Hypothesis,Phys. Rev. X9(2019) 041017 [1812.08657]

    work page arXiv 2019

  51. [51]

    Barb´ on, E

    J.L.F. Barb´ on, E. Rabinovici, R. Shir and R. Sinha,On The Evolution Of Operator Complexity Beyond Scrambling,JHEP10(2019) 264 [1907.05393]

    work page arXiv 2019

  52. [52]

    Hochbruck and C

    M. Hochbruck and C. Lubich,On krylov subspace approximations to the matrix exponential operator,SIAM Journal on Numerical Analysis34(1997) 1911 [https://doi.org/10.1137/S0036142995280572]

    work page doi:10.1137/s0036142995280572 1997

  53. [53]

    Vaccaro and D.T

    J.A. Vaccaro and D.T. Pegg,Wigner function for number and phase,Phys. Rev. A41(1990) 5156

    1990

  54. [54]

    Lukˇ s and V

    A. Lukˇ s and V. Peˇ rinov´ a,Ordering of ”ladder” operators, the wigner function for number and phase, and the enlarged hilbert space,Physica Scripta1993(1993) 94

    1993

  55. [55]

    Hush, A.R.R

    M.R. Hush, A.R.R. Carvalho and J.J. Hope,Number-phase wigner representation for efficient stochastic simulations,Phys. Rev. A81(2010) 033852

    2010

  56. [56]

    Mukunda,Wigner distribution for angle coordinates in quantum mechanics,American Journal of Physics47(1979) 182

    N. Mukunda,Wigner distribution for angle coordinates in quantum mechanics,American Journal of Physics47(1979) 182

    1979

  57. [57]

    Bizarro,Weyl-wigner formalism for rotation-angle and angular-momentum variables in quantum mechanics,Phys

    J.a.P. Bizarro,Weyl-wigner formalism for rotation-angle and angular-momentum variables in quantum mechanics,Phys. Rev. A49(1994) 3255

    1994

  58. [58]

    Mari and J

    A. Mari and J. Eisert,Positive wigner functions render classical simulation of quantum computation efficient,Physical Review Letters109(2012)

    2012

  59. [59]

    Pashayan, J.J

    H. Pashayan, J.J. Wallman and S.D. Bartlett,Estimating outcome probabilities of quantum circuits using quasiprobabilities,Physical Review Letters115(2015)

    2015

  60. [60]

    Quantum Entanglement of Local Operators in Conformal Field Theories

    M. Nozaki, T. Numasawa and T. Takayanagi,Quantum Entanglement of Local Operators in Conformal Field Theories,Phys. Rev. Lett.112(2014) 111602 [1401.0539]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  61. [61]

    Caputa, J.M

    P. Caputa, J.M. Magan, D. Patramanis and E. Tonni,Krylov complexity of modular Hamiltonian evolution,Phys. Rev. D109(2024) 086004 [2306.14732]

    work page arXiv 2024

  62. [62]

    Caputa and G

    P. Caputa and G. Di Giulio,Local quenches from a Krylov perspective,JHEP07(2025) 164 [2502.19485]

    work page arXiv 2025

  63. [63]

    Caputa, B

    P. Caputa, B. Chen, R.W. McDonald, J. Sim´ on and B. Strittmatter,Spread complexity rate as proper momentum,Phys. Rev. D113(2026) L041901 [2410.23334]

    work page arXiv 2026

  64. [64]

    Holographic Local Quenches and Entanglement Density

    M. Nozaki, T. Numasawa and T. Takayanagi,Holographic Local Quenches and Entanglement Density,JHEP05(2013) 080 [1302.5703]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  65. [65]

    Balasubramanian and S.F

    V. Balasubramanian and S.F. Ross,Holographic particle detection,Physical Review D61 (2000) 044007

    2000

  66. [66]

    Dumitriu and A

    I. Dumitriu and A. Edelman,Matrix models for beta ensembles,Journal of Mathematical Physics43(2002) 5830 [https://pubs.aip.org/aip/jmp/article-pdf/43/11/5830/19137139/5830 1 online.pdf]. – 42 –

    2002

  67. [67]

    Susskind and J

    L. Susskind and J. Glogower,Quantum mechanical phase and time operator,Physics Physique Fizika1(1964) 49

    1964

  68. [68]

    Carruthers and M.M.M

    P. Carruthers and M.M.M. Nieto,Phase and angle variables in quantum mechanics,Rev. Mod. Phys.40(1968) 411

    1968

  69. [69]

    White and J.H

    C.D. White and J.H. Wilson,Mana in Haar-random states,2011.13937

    work page arXiv 2011

  70. [70]

    Black Holes and Random Matrices

    J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker et al.,Black Holes and Random Matrices,JHEP05(2017) 118 [1611.04650]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  71. [71]

    Caputa, J.M

    P. Caputa, J.M. Magan and D. Patramanis,Geometry of Krylov complexity,Phys. Rev. Res. 4(2022) 013041 [2109.03824]

    work page arXiv 2022

  72. [72]

    State-Dependent Bulk-Boundary Maps and Black Hole Complementarity

    K. Papadodimas and S. Raju,State-Dependent Bulk-Boundary Maps and Black Hole Complementarity,Phys. Rev. D89(2014) 086010 [1310.6335]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  73. [73]

    Bulk Locality and Quantum Error Correction in AdS/CFT

    A. Almheiri, X. Dong and D. Harlow,Bulk Locality and Quantum Error Correction in AdS/CFT,JHEP04(2015) 163 [1411.7041]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  74. [74]

    Quantum Computation vs. Firewalls

    D. Harlow and P. Hayden,Quantum Computation vs. Firewalls,JHEP06(2013) 085 [1301.4504]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  75. [75]

    Susskind,Computational complexity and black hole horizons, 2014

    L. Susskind,Computational complexity and black hole horizons, 2014

    2014

  76. [76]

    Entanglement is not Enough

    L. Susskind,Entanglement is not enough,Fortsch. Phys.64(2016) 49 [1411.0690]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  77. [77]

    The Typical-State Paradox: Diagnosing Horizons with Complexity

    L. Susskind,The Typical-State Paradox: Diagnosing Horizons with Complexity,Fortsch. Phys.64(2016) 84 [1507.02287]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  78. [78]

    Brown, H

    A.R. Brown, H. Gharibyan, G. Penington and L. Susskind,The Python’s Lunch: geometric obstructions to decoding Hawking radiation,JHEP08(2020) 121 [1912.00228]

    work page arXiv 2020

  79. [79]

    I.H. Kim, E. Tang and J. Preskill,The ghost in the radiation: robust encodings of the black hole interior (invited paper),JHEP06(2020) 031 [2003.05451]

    work page arXiv 2020

  80. [80]

    Balasubramanian, A

    V. Balasubramanian, A. Kar, C. Li and O. Parrikar,Quantum error correction in the black hole interior,JHEP07(2023) 189 [2203.01961]

    work page arXiv 2023

Showing first 80 references.