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More on complexity of operators in quantum field theory

  • Run-Qiu Yang
  • Yu-Sen An
  • Chao Niu
  • Cheng-Yong Zhang
  • Keun-Young KimEmail author
Open Access
Regular Article - Theoretical Physics
  • 50 Downloads

Abstract

Recently it has been shown that the complexity of SU(n) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten p-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as k-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU(n) groups.

Keywords

Gauge-gravity correspondence Holography and condensed matter physics (AdS/CMT) 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    S. Aaronson, The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes, arXiv:1607.05256 [INSPIRE].
  2. [2]
    D. Harlow and P. Hayden, Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
  3. [3]
    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].ADSGoogle Scholar
  6. [6]
    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang and R.-H. Peng, Action growth for AdS black holes, JHEP 09 (2016) 161 [arXiv:1606.08307] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Reynolds and S.F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R.-Q. Yang, C. Niu and K.-Y. Kim, Surface Counterterms and Regularized Holographic Complexity, JHEP 09 (2017) 042 [arXiv:1701.03706] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the Time Dependence of Holographic Complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R.-Q. Yang, C. Niu, C.-Y. Zhang and K.-Y. Kim, Comparison of holographic and field theoretic complexities for time dependent thermofield double states, JHEP 02 (2018) 082 [arXiv:1710.00600] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    B. Swingle and Y. Wang, Holographic Complexity of Einstein-Maxwell-Dilaton Gravity, JHEP 09 (2018) 106 [arXiv:1712.09826] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
  19. [19]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
  20. [20]
    Y.-S. An, R.-G. Cai and Y. Peng, Time Dependence of Holographic Complexity in Gauss-Bonnet Gravity, Phys. Rev. D 98 (2018) 106013 [arXiv:1805.07775] [INSPIRE].ADSGoogle Scholar
  21. [21]
    K. Nagasaki, Complexity growth of rotating black holes with a probe string, Phys. Rev. D 98 (2018) 126014 [arXiv:1807.01088] [INSPIRE].ADSGoogle Scholar
  22. [22]
    S. Mahapatra and P. Roy, On the time dependence of holographic complexity in a dynamical Einstein-dilaton model, JHEP 11 (2018) 138 [arXiv:1808.09917] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    O. Ben-Ami and D. Carmi, On Volumes of Subregions in Holography and Complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    B. Chen, W.-M. Li, R.-Q. Yang, C.-Y. Zhang and S.-J. Zhang, Holographic subregion complexity under a thermal quench, JHEP 07 (2018) 034 [arXiv:1803.06680] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Y. Ling, Y. Liu and C.-Y. Zhang, Holographic Subregion Complexity in Einstein-Born-Infeld theory, Eur. Phys. J. C 79 (2019) 194 [arXiv:1808.10169] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, JHEP 02 (2019) 145 [arXiv:1804.01561] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
  30. [30]
    A.R. Brown, L. Susskind and Y. Zhao, Quantum Complexity and Negative Curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton, Class. Quant. Grav. 35 (2018) 095006 [arXiv:1712.03732] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, Phys. Rev. D 98 (2018) 126001 [arXiv:1801.07620] [INSPIRE].ADSGoogle Scholar
  35. [35]
    L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit Complexity for Coherent States, JHEP 10 (2018) 011 [arXiv:1807.07677] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Bhattacharyya, A. Shekar and A. Sinha, Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [arXiv:1808.03105] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    H.A. Camargo, P. Caputa, D. Das, M.P. Heller and R. Jefferson, Complexity as a novel probe of quantum quenches: universal scalings and purifications, Phys. Rev. Lett. 122 (2019) 081601 [arXiv:1807.07075] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a Definition of Complexity for Quantum Field Theory States, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    T. Takayanagi, Holographic Spacetimes as Quantum Circuits of Path-Integrations, JHEP 12 (2018) 048 [arXiv:1808.09072] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    K. Hashimoto, N. Iizuka and S. Sugishita, Time evolution of complexity in Abelian gauge theories, Phys. Rev. D 96 (2017) 126001 [arXiv:1707.03840] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    K. Hashimoto, N. Iizuka and S. Sugishita, Thoughts on Holographic Complexity and its Basis-dependence, Phys. Rev. D 98 (2018) 046002 [arXiv:1805.04226] [INSPIRE].ADSGoogle Scholar
  45. [45]
    J.M. Magán, Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [arXiv:1805.05839] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    P. Caputa and J.M. Magan, Quantum Computation as Gravity, arXiv:1807.04422 [INSPIRE].
  47. [47]
    M. Flory and N. Miekley, Complexity change under conformal transformations in AdS 3 /CFT 2, arXiv:1806.08376 [INSPIRE].
  48. [48]
    R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].ADSMathSciNetGoogle Scholar
  49. [49]
    R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, Principles and symmetries of complexity in quantum field theory, Eur. Phys. J. C 79 (2019) 109 [arXiv:1803.01797] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum computation as geometry, Science 311 (2006) 1133.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Inf. Comput. 6 (2006) 213.MathSciNetzbMATHGoogle Scholar
  52. [52]
    M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, Quant. Inf. Comput. 8 (2008) 861.MathSciNetzbMATHGoogle Scholar
  53. [53]
    J. Watrous, Theory of quantum information, Cambridge University Press (2018).Google Scholar
  54. [54]
    D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer New York, New York (2000) [DOI: https://doi.org/10.1007/978-1-4612-1268-3].CrossRefzbMATHGoogle Scholar
  55. [55]
    Z. Shen, Lectures on Finsler geometry, Series on Multivariate Analysis, Wspc (2001).Google Scholar
  56. [56]
    M. Xiaohuan, An introduction to Finsler geometry, World Scientific, Singapore Hackensack, NJ (2006).zbMATHGoogle Scholar
  57. [57]
    G.S. Asanov, Finsler Geometry, Relativity and Gauge Theories, Springer Netherlands, Dordrecht (1985).CrossRefzbMATHGoogle Scholar
  58. [58]
    D. Latifi and M. Toomanian, On the existence of bi-invariant finsler metrics on lie groups, Math. Sci. 7 (2013) 37.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    D. Latifi and A. Razavi, Bi-invariant finsler metrics on lie groups, J. Basic Appl. Sci. 5 (2011) 607.Google Scholar
  60. [60]
    R.-Q. Yang and K.-Y. Kim, Complexity of operators generated by quantum mechanical Hamiltonians, JHEP 03 (2019) 010 [arXiv:1810.09405] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    M.M. Alexandrino and R.G. Bettiol, Lie Groups with Bi-invariant Metrics, Springer International Publishing, Cham (2015), pp. 27–47.Google Scholar
  62. [62]
    S. Deng and Z. Hou, Positive definite minkowski lie algebras and bi-invariant finsler metrics on lie groups, Geom. Dedicata 136 (2008) 191.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    T. Frankel, The geometry of physics: an introduction, Cambridge University Press, Cambridge, New York (2012).zbMATHGoogle Scholar
  64. [64]
    R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, Complexity between states in quantum field theories, work in progress.Google Scholar
  65. [65]
    C. Zachos, D. Fairlie and T. Curtright, Quantum Mechanics in Phase Space: An Overview with Selected Papers, World Scientific (2005).Google Scholar
  66. [66]
    T.L. Curtright and C.K. Zachos, Quantum Mechanics in Phase Space, Asia Pac. Phys. Newslett. 1 (2012) 37 [arXiv:1104.5269] [INSPIRE].CrossRefGoogle Scholar
  67. [67]
    E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  68. [68]
    S.S. Chern, Lectures on differential geometry, World Scientific, Singapore River Edge, NJ (1999).CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Run-Qiu Yang
    • 1
  • Yu-Sen An
    • 2
    • 3
  • Chao Niu
    • 4
  • Cheng-Yong Zhang
    • 5
  • Keun-Young Kim
    • 6
    Email author
  1. 1.Quantum Universe CenterKorea Institute for Advanced StudySeoulKorea
  2. 2.Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of ScienceBeijingChina
  3. 3.School of physical ScienceUniversity of Chinese Academy of ScienceBeijingChina
  4. 4.Department of Physics and Siyuan LaboratoryJinan UniversityGuangzhouChina
  5. 5.Department of Physics and Center for Field Theory and Particle PhysicsFudan UniversityShanghaiChina
  6. 6.School of Physics and ChemistryGwangju Institute of Science and TechnologyGwangjuKorea

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