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.
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References
S. Aaronson, The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes, arXiv:1607.05256 [INSPIRE].
D. Harlow and P. Hayden, Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
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].
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].
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].
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].
S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].
D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].
A. Reynolds and S.F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].
R.-Q. Yang, C. Niu and K.-Y. Kim, Surface Counterterms and Regularized Holographic Complexity, JHEP 09 (2017) 042 [arXiv:1701.03706] [INSPIRE].
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].
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].
B. Swingle and Y. Wang, Holographic Complexity of Einstein-Maxwell-Dilaton Gravity, JHEP 09 (2018) 106 [arXiv:1712.09826] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
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].
K. Nagasaki, Complexity growth of rotating black holes with a probe string, Phys. Rev. D 98 (2018) 126014 [arXiv:1807.01088] [INSPIRE].
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].
M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
O. Ben-Ami and D. Carmi, On Volumes of Subregions in Holography and Complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].
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].
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].
C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, JHEP 02 (2019) 145 [arXiv:1804.01561] [INSPIRE].
J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].
L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
A.R. Brown, L. Susskind and Y. Zhao, Quantum Complexity and Negative Curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].
A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton, Class. Quant. Grav. 35 (2018) 095006 [arXiv:1712.03732] [INSPIRE].
R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, Phys. Rev. D 98 (2018) 126001 [arXiv:1801.07620] [INSPIRE].
L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].
M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit Complexity for Coherent States, JHEP 10 (2018) 011 [arXiv:1807.07677] [INSPIRE].
A. Bhattacharyya, A. Shekar and A. Sinha, Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [arXiv:1808.03105] [INSPIRE].
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].
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].
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].
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].
T. Takayanagi, Holographic Spacetimes as Quantum Circuits of Path-Integrations, JHEP 12 (2018) 048 [arXiv:1808.09072] [INSPIRE].
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].
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].
J.M. Magán, Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [arXiv:1805.05839] [INSPIRE].
P. Caputa and J.M. Magan, Quantum Computation as Gravity, arXiv:1807.04422 [INSPIRE].
M. Flory and N. Miekley, Complexity change under conformal transformations in AdS 3 /CFT 2, arXiv:1806.08376 [INSPIRE].
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].
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].
M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum computation as geometry, Science 311 (2006) 1133.
M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Inf. Comput. 6 (2006) 213.
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, Quant. Inf. Comput. 8 (2008) 861.
J. Watrous, Theory of quantum information, Cambridge University Press (2018).
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].
Z. Shen, Lectures on Finsler geometry, Series on Multivariate Analysis, Wspc (2001).
M. Xiaohuan, An introduction to Finsler geometry, World Scientific, Singapore Hackensack, NJ (2006).
G.S. Asanov, Finsler Geometry, Relativity and Gauge Theories, Springer Netherlands, Dordrecht (1985).
D. Latifi and M. Toomanian, On the existence of bi-invariant finsler metrics on lie groups, Math. Sci. 7 (2013) 37.
D. Latifi and A. Razavi, Bi-invariant finsler metrics on lie groups, J. Basic Appl. Sci. 5 (2011) 607.
R.-Q. Yang and K.-Y. Kim, Complexity of operators generated by quantum mechanical Hamiltonians, JHEP 03 (2019) 010 [arXiv:1810.09405] [INSPIRE].
M.M. Alexandrino and R.G. Bettiol, Lie Groups with Bi-invariant Metrics, Springer International Publishing, Cham (2015), pp. 27–47.
S. Deng and Z. Hou, Positive definite minkowski lie algebras and bi-invariant finsler metrics on lie groups, Geom. Dedicata 136 (2008) 191.
T. Frankel, The geometry of physics: an introduction, Cambridge University Press, Cambridge, New York (2012).
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.
C. Zachos, D. Fairlie and T. Curtright, Quantum Mechanics in Phase Space: An Overview with Selected Papers, World Scientific (2005).
T.L. Curtright and C.K. Zachos, Quantum Mechanics in Phase Space, Asia Pac. Phys. Newslett. 1 (2012) 37 [arXiv:1104.5269] [INSPIRE].
E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749 [INSPIRE].
S.S. Chern, Lectures on differential geometry, World Scientific, Singapore River Edge, NJ (1999).
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Yang, RQ., An, YS., Niu, C. et al. More on complexity of operators in quantum field theory. J. High Energ. Phys. 2019, 161 (2019). https://doi.org/10.1007/JHEP03(2019)161
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DOI: https://doi.org/10.1007/JHEP03(2019)161