1Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, G. D. Luxembourg
2Department of Physics Engineering, Faculty of Engineering, Mie University, Mie 514–8507, Japan
3Donostia International Physics Center, E-20018 San Sebastián, Spain
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Abstract
Quantum speed limits (QSLs) provide lower bounds on the minimum time required for a process to unfold by using a distance between quantum states and identifying the speed of evolution or an upper bound to it. We introduce a generalization of QSL to characterize the evolution of a general operator when conjugated by a unitary. The resulting operator QSL (OQSL) admits a geometric interpretation, is shown to be tight, and holds for operator flows induced by arbitrary unitaries, i.e., with time- or parameter-dependent generators. The derived OQSL is applied to the Wegner flow equations in Hamiltonian renormalization group theory and the operator growth quantified by the Krylov complexity.
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[1] P. Pfeifer and J. Fröhlich, “Generalized time-energy uncertainty relations and bounds on lifetimes of resonances,” Rev. Mod. Phys. 67, 759–779 (1995).
https://doi.org/10.1103/RevModPhys.67.759
[2] P. Busch, “The time–energy uncertainty relation,” in Time in Quantum Mechanics, edited by J. G. Muga, R. S. Mayato, and Í. L. Egusquiza (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) pp. 73–105.
https://doi.org/10.1007/978-3-540-73473-4_3
[3] L. S. Schulman, “Jump time and passage time: The duration ofs a quantum transition,” in Time in Quantum Mechanics, edited by J. G. Muga, R. Sala Mayato, and Í. L. Egusquiza (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) pp. 107–128.
https://doi.org/10.1007/978-3-540-73473-4_4
[4] V. V. Dodonov and A. V. Dodonov, “Energy–time and frequency–time uncertainty relations: exact inequalities,” Physica Scripta 90, 074049 (2015).
https://doi.org/10.1088/0031-8949/90/7/074049
[5] L. Mandelstam and I. Tamm, “The uncertainty relation between energy and time in non-relativistic quantum mechanics,” in Selected Papers, edited by Boris M. Bolotovskii, Victor Ya. Frenkel, and Rudolf Peierls (Springer Berlin Heidelberg, Berlin, Heidelberg, 1991) pp. 115–123.
https://doi.org/10.1007/978-3-642-74626-0_8
[6] S. Deffner and S. Campbell, “Quantum speed limits: from heisenberg’s uncertainty principle to optimal quantum control,” Journal of Physics A: Mathematical and Theoretical 50, 453001 (2017).
https://doi.org/10.1088/1751-8121/aa86c6
[7] Z. Gong and R. Hamazaki, “Bounds in nonequilibrium quantum dynamics,” International Journal of Modern Physics B 36, 2230007 (2022).
https://doi.org/10.1142/S0217979222300079
[8] N. Margolus and L. B. Levitin, “The maximum speed of dynamical evolution,” Physica D: Nonlinear Phenomena 120, 188–195 (1998), proceedings of the Fourth Workshop on Physics and Consumption.
https://doi.org/10.1016/S0167-2789(98)00054-2
[9] B. Zieliński and M. Zych, “Generalization of the margolus-levitin bound,” Phys. Rev. A 74, 034301 (2006).
https://doi.org/10.1103/PhysRevA.74.034301
[10] N. Margolus, “The finite-state character of physical dynamics,” arXiv e-prints , arXiv:1109.4994 (2011), arXiv:1109.4994 [quant-ph].
arXiv:1109.4994
[11] A. Uhlmann, “An energy dispersion estimate,” Physics Letters A 161, 329–331 (1992).
https://doi.org/10.1016/0375-9601(92)90555-Z
[12] S. Deffner and E. Lutz, “Energy–time uncertainty relation for driven quantum systems,” Journal of Physics A: Mathematical and Theoretical 46, 335302 (2013a).
https://doi.org/10.1088/1751-8113/46/33/335302
[13] M. Okuyama and M. Ohzeki, “Comment on `energy-time uncertainty relation for driven quantum systems’,” Journal of Physics A: Mathematical and Theoretical 51, 318001 (2018a).
https://doi.org/10.1088/1751-8121/aacb90
[14] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, “Quantum speed limit for physical processes,” Phys. Rev. Lett. 110, 050402 (2013).
https://doi.org/10.1103/PhysRevLett.110.050402
[15] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, “Quantum speed limits in open system dynamics,” Phys. Rev. Lett. 110, 050403 (2013).
https://doi.org/10.1103/PhysRevLett.110.050403
[16] S. Deffner and E. Lutz, “Quantum speed limit for non-markovian dynamics,” Phys. Rev. Lett. 111, 010402 (2013b).
https://doi.org/10.1103/PhysRevLett.111.010402
[17] F. Campaioli, F. A. Pollock, and K. Modi, “Tight, robust, and feasible quantum speed limits for open dynamics,” Quantum 3, 168 (2019).
https://doi.org/10.22331/q-2019-08-05-168
[18] L. P. García-Pintos and A. del Campo, “Quantum speed limits under continuous quantum measurements,” New Journal of Physics 21, 033012 (2019).
https://doi.org/10.1088/1367-2630/ab099e
[19] B. Shanahan, A. Chenu, N. Margolus, and A. del Campo, “Quantum speed limits across the quantum-to-classical transition,” Phys. Rev. Lett. 120, 070401 (2018).
https://doi.org/10.1103/PhysRevLett.120.070401
[20] M. Okuyama and M. Ohzeki, “Quantum speed limit is not quantum,” Phys. Rev. Lett. 120, 070402 (2018b).
https://doi.org/10.1103/PhysRevLett.120.070402
[21] N. Shiraishi, K. Funo, and K. Saito, “Speed limit for classical stochastic processes,” Phys. Rev. Lett. 121, 070601 (2018).
https://doi.org/10.1103/PhysRevLett.121.070601
[22] S. B. Nicholson, L. P. García-Pintos, A. del Campo, and J. R. Green, “Time–information uncertainty relations in thermodynamics,” Nature Physics 16, 1211–1215 (2020).
https://doi.org/10.1038/s41567-020-0981-y
[23] V. T. Vo, T. Van Vu, and Y. Hasegawa, “Unified approach to classical speed limit and thermodynamic uncertainty relation,” Phys. Rev. E 102, 062132 (2020).
https://doi.org/10.1103/PhysRevE.102.062132
[24] T. Van Vu and Y. Hasegawa, “Geometrical bounds of the irreversibility in markovian systems,” Phys. Rev. Lett. 126, 010601 (2021).
https://doi.org/10.1103/PhysRevLett.126.010601
[25] L. P. García-Pintos, S. B. Nicholson, J. R. Green, A. del Campo, and A. V. Gorshkov, “Unifying quantum and classical speed limits on observables,” Phys. Rev. X 12, 011038 (2022).
https://doi.org/10.1103/PhysRevX.12.011038
[26] I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd ed. (Cambridge University Press, 2017).
https://doi.org/10.1017/9781139207010
[27] D. P. Pires, M. Cianciaruso, L. C. Céleri, G. Adesso, and D. O. Soares-Pinto, “Generalized geometric quantum speed limits,” Phys. Rev. X 6, 021031 (2016).
https://doi.org/10.1103/PhysRevX.6.021031
[28] F. Campaioli, F. A. Pollock, F. C. Binder, and K. Modi, “Tightening quantum speed limits for almost all states,” Phys. Rev. Lett. 120, 060409 (2018).
https://doi.org/10.1103/PhysRevLett.120.060409
[29] N. Hörnedal, D. Allan, and O. Sönnerborn, “Extensions of the mandelstam–tamm quantum speed limit to systems in mixed states,” New Journal of Physics 24, 055004 (2022a).
https://doi.org/10.1088/1367-2630/ac688a
[30] M. Bukov, D. Sels, and A. Polkovnikov, “Geometric speed limit of accessible many-body state preparation,” Phys. Rev. X 9, 011034 (2019).
https://doi.org/10.1103/PhysRevX.9.011034
[31] T. Fogarty, S. Deffner, T. Busch, and S. Campbell, “Orthogonality catastrophe as a consequence of the quantum speed limit,” Phys. Rev. Lett. 124, 110601 (2020).
https://doi.org/10.1103/PhysRevLett.124.110601
[32] K. Suzuki and K. Takahashi, “Performance evaluation of adiabatic quantum computation via quantum speed limits and possible applications to many-body systems,” Phys. Rev. Research 2, 032016 (2020).
https://doi.org/10.1103/PhysRevResearch.2.032016
[33] A. del Campo, “Probing quantum speed limits with ultracold gases,” Phys. Rev. Lett. 126, 180603 (2021).
https://doi.org/10.1103/PhysRevLett.126.180603
[34] R. Hamazaki, “Speed limits for macroscopic transitions,” PRX Quantum 3, 020319 (2022).
https://doi.org/10.1103/PRXQuantum.3.020319
[35] S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: Theory, examples, and lorentz invariance,” Annals of Physics 247, 135–173 (1996).
https://doi.org/10.1006/aphy.1996.0040
[36] V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nature Photonics 5, 222–229 (2011).
https://doi.org/10.1038/nphoton.2011.35
[37] G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).
https://doi.org/10.1088/1751-8113/47/42/424006
[38] M. Beau and A. del Campo, “Nonlinear quantum metrology of many-body open systems,” Phys. Rev. Lett. 119, 010403 (2017).
https://doi.org/10.1103/PhysRevLett.119.010403
[39] T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, and G. E. Santoro, “Optimal control at the quantum speed limit,” Phys. Rev. Lett. 103, 240501 (2009).
https://doi.org/10.1103/PhysRevLett.103.240501
[40] S. An, D. Lv, A. del Campo, and K. Kim, “Shortcuts to adiabaticity by counterdiabatic driving for trapped-ion displacement in phase space,” Nature Communications 7, 12999 (2016).
https://doi.org/10.1038/ncomms12999
[41] K. Funo, J.-N. Zhang, C. Chatou, K. Kim, M. Ueda, and A. del Campo, “Universal work fluctuations during shortcuts to adiabaticity by counterdiabatic driving,” Phys. Rev. Lett. 118, 100602 (2017).
https://doi.org/10.1103/PhysRevLett.118.100602
[42] S. Campbell and S. Deffner, “Trade-off between speed and cost in shortcuts to adiabaticity,” Phys. Rev. Lett. 118, 100601 (2017).
https://doi.org/10.1103/PhysRevLett.118.100601
[43] A. del Campo, J. Goold, and M. Paternostro, “More bang for your buck: Super-adiabatic quantum engines,” Scientific Reports 4, 6208 (2014).
https://doi.org/10.1038/srep06208
[44] F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, “Quantacell: powerful charging of quantum batteries,” New Journal of Physics 17, 075015 (2015).
https://doi.org/10.1088/1367-2630/17/7/075015
[45] F. Wegner, “Flow-equations for hamiltonians,” Annalen der Physik 506, 77–91 (1994).
https://doi.org/10.1002/andp.19945060203
[46] S. D. Głazek and K. G. Wilson, “Renormalization of hamiltonians,” Phys. Rev. D 48, 5863–5872 (1993).
https://doi.org/10.1103/PhysRevD.48.5863
[47] S. D. Glazek and K. G. Wilson, “Perturbative renormalization group for hamiltonians,” Phys. Rev. D 49, 4214–4218 (1994).
https://doi.org/10.1103/PhysRevD.49.4214
[48] F. J. Wegner, “Flow equations for hamiltonians,” Physics Reports 348, 77–89 (2001).
https://doi.org/10.1016/S0370-1573(00)00136-8
[49] S. Kehrein, The Flow Equation Approach to Many-Particle Systems, Springer Tracts in Modern Physics (Springer Berlin Heidelberg, 2007).
https://doi.org/10.1007/3-540-34068-8
[50] C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, “Operator hydrodynamics, otocs, and entanglement growth in systems without conservation laws,” Phys. Rev. X 8, 021013 (2018).
https://doi.org/10.1103/PhysRevX.8.021013
[51] A. Nahum, S. Vijay, and J. Haah, “Operator spreading in random unitary circuits,” Phys. Rev. X 8, 021014 (2018).
https://doi.org/10.1103/PhysRevX.8.021014
[52] T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, “Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation,” Phys. Rev. X 8, 031058 (2018).
https://doi.org/10.1103/PhysRevX.8.031058
[53] V. Khemani, A. Vishwanath, and D. A. Huse, “Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws,” Phys. Rev. X 8, 031057 (2018).
https://doi.org/10.1103/PhysRevX.8.031057
[54] D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, “A universal operator growth hypothesis,” Phys. Rev. X 9, 041017 (2019).
https://doi.org/10.1103/PhysRevX.9.041017
[55] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, And Correlation Functions, Advanced Books Classics (CRC Press, 2018).
[56] Nicoletta Carabba, Niklas Hörnedal, and Adolfo del Campo, “Quantum speed limits on operator flows and correlation functions,” Quantum 6, 884 (2022).
https://doi.org/10.22331/q-2022-12-22-884
[57] B. Mohan and A. K. Pati, “Quantum speed limits for observables,” Phys. Rev. A 106, 042436 (2022).
https://doi.org/10.1103/PhysRevA.106.042436
[58] J. L. F. Barbón, E. Rabinovici, R. Shir, and R. Sinha, “On the evolution of operator complexity beyond scrambling,” Journal of High Energy Physics 2019, 264 (2019).
https://doi.org/10.1007/JHEP10(2019)264
[59] P. Caputa, J. M. Magan, and D. Patramanis, “Geometry of krylov complexity,” Phys. Rev. Research 4, 013041 (2022).
https://doi.org/10.1103/PhysRevResearch.4.013041
[60] Anatoly Dymarsky and Alexander Gorsky, “Quantum chaos as delocalization in krylov space,” Phys. Rev. B 102, 085137 (2020).
https://doi.org/10.1103/PhysRevB.102.085137
[61] E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, “Operator complexity: a journey to the edge of krylov space,” Journal of High Energy Physics 2021, 62 (2021).
https://doi.org/10.1007/JHEP06(2021)062
[62] N. Hörnedal, N. Carabba, A. S. Matsoukas-Roubeas, and A. del Campo, “Ultimate speed limits to the growth of operator complexity,” Communications Physics 5, 207 (2022b).
https://doi.org/10.1038/s42005-022-00985-1
[63] H. Mori, “A Continued-Fraction Representation of the Time-Correlation Functions,” Progress of Theoretical Physics 34, 399–416 (1965).
https://doi.org/10.1143/PTP.34.399
[64] R. Kubo, “The fluctuation-dissipation theorem,” Reports on Progress in Physics 29, 255 (1966).
https://doi.org/10.1088/0034-4885/29/1/306
[65] G. Müller V. S. Viswanath, The Recursion Method: Application to Many-Body Dynamics (Springer-Verlag, 1994).
https://doi.org/10.1007/978-3-540-48651-0
[66] R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, International series of monographs on chemistry (Clarendon Press, 1990).
https://global.oup.com/academic/product/principles-of-nuclear-magnetic-resonance-in-one-and-two-dimensions-9780198556473
[67] J. A. Gyamfi, “Fundamentals of quantum mechanics in liouville space,” European Journal of Physics 41, 063002 (2020).
https://doi.org/10.1088/1361-6404/ab9fdd
[68] S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra (Pearson, 2019).
https://books.google.lu/books?id=zhw6vQEACAAJ
[69] L. B. Levitin and T. Toffoli, “Fundamental limit on the rate of quantum dynamics: The unified bound is tight,” Phys. Rev. Lett. 103, 160502 (2009).
https://doi.org/10.1103/PhysRevLett.103.160502
[70] J. Haegeman, T. J. Osborne, H. Verschelde, and F. Verstraete, “Entanglement renormalization for quantum fields in real space,” Phys. Rev. Lett. 110, 100402 (2013).
https://doi.org/10.1103/PhysRevLett.110.100402
[71] M. Nozaki, S. Ryu, and T. Takayanagi, “Holographic geometry of entanglement renormalization in quantum field theories,” Journal of High Energy Physics 2012, 193 (2012).
https://doi.org/10.1007/JHEP10(2012)193
[72] J. Molina-Vilaplana and A. del Campo, “Complexity functionals and complexity growth limits in continuous mera circuits,” Journal of High Energy Physics 2018, 12 (2018).
https://doi.org/10.1007/JHEP08(2018)012
[73] 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, 071602 (2017a).
https://doi.org/10.1103/PhysRevLett.119.071602
[74] P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, and K. Watanabe, “Liouville action as path-integral complexity: from continuous tensor networks to ads/cft,” Journal of High Energy Physics 2017, 97 (2017b).
https://doi.org/10.1007/JHEP11(2017)097
[75] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, “Complexity, action, and black holes,” Phys. Rev. D 93, 086006 (2016a).
https://doi.org/10.1103/PhysRevD.93.086006
[76] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, “Holographic complexity equals bulk action?” Phys. Rev. Lett. 116, 191301 (2016b).
https://doi.org/10.1103/PhysRevLett.116.191301
[77] R. Uzdin and R. Kosloff, “Speed limits in liouville space for open quantum systems,” EPL (Europhysics Letters) 115, 40003 (2016).
https://doi.org/10.1209/0295-5075/115/40003
[78] D. A. Lidar, A. Shabani, and R. Alicki, “Conditions for strictly purity-decreasing quantum markovian dynamics,” Chemical Physics 322, 82–86 (2006).
https://doi.org/10.1016/j.chemphys.2005.06.038
[79] M. Toda, “Vibration of a chain with nonlinear interaction,” Journal of the Physical Society of Japan 22, 431–436 (1967a).
https://doi.org/10.1143/JPSJ.22.431
[80] M. Toda, “Wave propagation in anharmonic lattices,” Journal of the Physical Society of Japan 23, 501–506 (1967b).
https://doi.org/10.1143/JPSJ.23.501
[81] H. Flaschka, “The toda lattice. ii. existence of integrals,” Phys. Rev. B 9, 1924–1925 (1974).
https://doi.org/10.1103/PhysRevB.9.1924
[82] J. Moser, Dynamical Systems, Theory and Applications (Springer, 1975).
https://doi.org/10.1007/3-540-07171-7
[83] C. Monthus, “Flow towards diagonalization for many-body-localization models: adaptation of the toda matrix differential flow to random quantum spin chains,” Journal of Physics A: Mathematical and Theoretical 49, 305002 (2016).
https://doi.org/10.1088/1751-8113/49/30/305002
[84] M. Okuyama and K. Takahashi, “From classical nonlinear integrable systems to quantum shortcuts to adiabaticity,” Phys. Rev. Lett. 117, 070401 (2016).
https://doi.org/10.1103/PhysRevLett.117.070401
[85] D. Chowdhury, A. Georges, O. Parcollet, and S. Sachdev, “Sachdev-ye-kitaev models and beyond: Window into non-fermi liquids,” Rev. Mod. Phys. 94, 035004 (2022).
https://doi.org/10.1103/RevModPhys.94.035004
[86] S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–wolff transformation for quantum many-body systems,” Annals of Physics 326, 2793–2826 (2011).
https://doi.org/10.1016/j.aop.2011.06.004
[87] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin Heidelberg, 2007).
https://doi.org/10.1007/978-3-540-69969-9
[88] B. Sutherland, Beautiful Models (World Scientific, 2004).
https://doi.org/10.1142/5552
[89] D. J. Gross, J. Kruthoff, A. Rolph, and E. Shaghoulian, “$toverline{T}$ in ${mathrm{ads}}_{2}$ and quantum mechanics,” Phys. Rev. D 101, 026011 (2020a).
https://doi.org/10.1103/PhysRevD.101.026011
[90] D. J. Gross, J. Kruthoff, A. Rolph, and E. Shaghoulian, “Hamiltonian deformations in quantum mechanics, $toverline{T}$, and the syk model,” Phys. Rev. D 102, 046019 (2020b).
https://doi.org/10.1103/PhysRevD.102.046019
[91] A. S. Matsoukas-Roubeas, F. Roccati, J. Cornelius, Z. Xu, A. Chenu, and A. del Campo, “Non-hermitian hamiltonian deformations in quantum mechanics,” (2022).
https://doi.org/10.48550/ARXIV.2211.05437
[92] Moody T. Chu and Kenneth R. Driessel, “The projected gradient method for least squares matrix approximations with spectral constraints,” SIAM Journal on Numerical Analysis 27, 1050–1060 (1990).
http://www.jstor.org/stable/2157698
[93] R. W. Brockett, “Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems,” Linear Algebra and its Applications 146, 79–91 (1991).
https://doi.org/10.1016/0024-3795(91)90021-N
[94] A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, “Operator growth and krylov construction in dissipative open quantum systems,” Journal of High Energy Physics 2022, 81 (2022).
https://doi.org/10.1007/JHEP12(2022)081
[95] C. Liu, H. Tang, and H. Zhai, “Krylov complexity in open quantum systems,” (2022).
https://doi.org/10.48550/ARXIV.2207.13603
[96] Budhaditya Bhattacharjee, Xiangyu Cao, Pratik Nandy, and Tanay Pathak, “An operator growth hypothesis for open quantum systems,” (2022).
https://doi.org/10.48550/ARXIV.2212.06180
Cited by
[1] Dimitrios Patramanis and Watse Sybesma, “Krylov complexity in a natural basis for the Schrödinger algebra”, arXiv:2306.03133, (2023).
[2] Ryusuke Hamazaki, “Quantum Velocity Limits for Multiple Observables: Conservation Laws, Correlations, and Macroscopic Systems”, arXiv:2305.03190, (2023).
[3] Pawel Caputa, Javier M. Magan, Dimitrios Patramanis, and Erik Tonni, “Krylov complexity of modular Hamiltonian evolution”, arXiv:2306.14732, (2023).
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