Abstract
Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity (CK) and Spread complexity (CS) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators. After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between CK and 2CS. Furthermore, for maximally entangled pure states, we find that the moment-generating function of CK becomes the Spectral Form Factor and, at late-times, CK is simply related to NCS for N ≥ 2 within the N-dimensional Hilbert space. Notably, we confirm that CK = 2CS holds across all times when N = 2. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. E. Shannon, The synthesis of two-terminal switching circuits, Bell Syst. Tech. J. 28 (1949) 59.
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, Quant. Inf. Comput. 8 (2008) 0861 [quant-ph/0701004] [INSPIRE].
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
P. Caputa et al., Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [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].
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, Phys. Rev. Lett. 122 (2019) 231302 [arXiv:1807.04422] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
A.R. Brown et al., Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
A.R. Brown et al., Complexity, action, and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume, and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].
A. Belin et al., Does Complexity Equal Anything?, Phys. Rev. Lett. 128 (2022) 081602 [arXiv:2111.02429] [INSPIRE].
D.E. Parker et al., A Universal Operator Growth Hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [INSPIRE].
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].
V. Balasubramanian, P. Caputa, J.M. Magan and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106 (2022) 046007 [arXiv:2202.06957] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Krylov complexity from integrability to chaos, JHEP 07 (2022) 151 [arXiv:2207.07701] [INSPIRE].
V. Balasubramanian, J.M. Magan and Q. Wu, Quantum chaos, integrability, and late times in the Krylov basis, arXiv:2312.03848 [INSPIRE].
P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B 106 (2022) 195125 [arXiv:2205.05688] [INSPIRE].
P. Caputa et al., Spread complexity and topological transitions in the Kitaev chain, JHEP 01 (2023) 120 [arXiv:2208.06311] [INSPIRE].
V. Balasubramanian, J.M. Magan and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107 (2023) 126001 [arXiv:2208.08452] [INSPIRE].
H.W. Lin, The bulk Hilbert space of double scaled SYK, JHEP 11 (2022) 060 [arXiv:2208.07032] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, A bulk manifestation of Krylov complexity, JHEP 08 (2023) 213 [arXiv:2305.04355] [INSPIRE].
J.L.F. Barbón, E. Rabinovici, R. Shir and R. Sinha, On The Evolution Of Operator Complexity Beyond Scrambling, JHEP 10 (2019) 264 [arXiv:1907.05393] [INSPIRE].
A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2 (2020) 043234 [arXiv:1911.09672] [INSPIRE].
A. Dymarsky and A. Gorsky, Quantum chaos as delocalization in Krylov space, Phys. Rev. B 102 (2020) 085137 [arXiv:1912.12227] [INSPIRE].
J.M. Magán and J. Simón, On operator growth and emergent Poincaré symmetries, JHEP 05 (2020) 071 [arXiv:2002.03865] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Operator complexity: a journey to the edge of Krylov space, JHEP 06 (2021) 062 [arXiv:2009.01862] [INSPIRE].
X. Cao, A statistical mechanism for operator growth, J. Phys. A 54 (2021) 144001 [arXiv:2012.06544] [INSPIRE].
S.-K. Jian, B. Swingle and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT gravity, JHEP 03 (2021) 014 [arXiv:2008.12274] [INSPIRE].
J. Kim, J. Murugan, J. Olle and D. Rosa, Operator delocalization in quantum networks, Phys. Rev. A 105 (2022) L010201 [arXiv:2109.05301] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Krylov localization and suppression of complexity, JHEP 03 (2022) 211 [arXiv:2112.12128] [INSPIRE].
A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104 (2021) L081702 [arXiv:2104.09514] [INSPIRE].
P. Caputa, J.M. Magan and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4 (2022) 013041 [arXiv:2109.03824] [INSPIRE].
D. Patramanis, Probing the entanglement of operator growth, PTEP 2022 (2022) 063A01 [arXiv:2111.03424] [INSPIRE].
P. Caputa and S. Datta, Operator growth in 2d CFT, JHEP 12 (2021) 188 [Erratum ibid. 09 (2022) 113] [arXiv:2110.10519] [INSPIRE].
F.B. Trigueros and C.-J. Lin, Krylov complexity of many-body localization: Operator localization in Krylov basis, SciPost Phys. 13 (2022) 037 [arXiv:2112.04722] [INSPIRE].
Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A 105 (2022) 062210 [arXiv:2202.07220] [INSPIRE].
R. Heveling, J. Wang and J. Gemmer, Numerically probing the universal operator growth hypothesis, Phys. Rev. E 106 (2022) 014152 [arXiv:2203.00533] [INSPIRE].
B. Bhattacharjee, X. Cao, P. Nandy and T. Pathak, Krylov complexity in saddle-dominated scrambling, JHEP 05 (2022) 174 [arXiv:2203.03534] [INSPIRE].
W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nucl. Phys. B 984 (2022) 115948 [arXiv:2205.12815] [INSPIRE].
S. He, P.H.C. Lau, Z.-Y. Xian and L. Zhao, Quantum chaos, scrambling and operator growth in \( T\overline{T} \) deformed SYK models, JHEP 12 (2022) 070 [arXiv:2209.14936] [INSPIRE].
N. Hörnedal, N. Carabba, A.S. Matsoukas-Roubeas and A. del Campo, Ultimate Speed Limits to the Growth of Operator Complexity, Commun. Phys. 5 (2022) 207 [arXiv:2202.05006] [INSPIRE].
M. Alishahiha and S. Banerjee, A universal approach to Krylov state and operator complexities, SciPost Phys. 15 (2023) 080 [arXiv:2212.10583] [INSPIRE].
H.A. Camargo, V. Jahnke, K.-Y. Kim and M. Nishida, Krylov complexity in free and interacting scalar field theories with bounded power spectrum, JHEP 05 (2023) 226 [arXiv:2212.14702] [INSPIRE].
S. Baek, Krylov complexity in inverted harmonic oscillator, arXiv:2210.06815 [INSPIRE].
A. Bhattacharya, P. Nandy, P.P. Nath and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, JHEP 12 (2022) 081 [arXiv:2207.05347] [INSPIRE].
C. Liu, H. Tang and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5 (2023) 033085 [arXiv:2207.13603] [INSPIRE].
B. Bhattacharjee, X. Cao, P. Nandy and T. Pathak, Operator growth in open quantum systems: lessons from the dissipative SYK, JHEP 03 (2023) 054 [arXiv:2212.06180] [INSPIRE].
M. Afrasiar et al., Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model, J. Stat. Mech. 2310 (2023) 103101 [arXiv:2208.10520] [INSPIRE].
A. Avdoshkin, A. Dymarsky and M. Smolkin, Krylov complexity in quantum field theory, and beyond, arXiv:2212.14429 [INSPIRE].
J. Erdmenger, S.-K. Jian and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, JHEP 08 (2023) 176 [arXiv:2303.12151] [INSPIRE].
K. Hashimoto, K. Murata, N. Tanahashi and R. Watanabe, Krylov complexity and chaos in quantum mechanics, JHEP 11 (2023) 040 [arXiv:2305.16669] [INSPIRE].
H.A. Camargo et al., Spectral and Krylov complexity in billiard systems, Phys. Rev. D 109 (2024) 046017 [arXiv:2306.11632] [INSPIRE].
A. Bhattacharya, R.N. Das, B. Dey and J. Erdmenger, Spread complexity for measurement-induced non-unitary dynamics and Zeno effect, JHEP 03 (2024) 179 [arXiv:2312.11635] [INSPIRE].
M.J. Vasli et al., Krylov complexity in Lifshitz-type scalar field theories, Eur. Phys. J. C 84 (2024) 235 [arXiv:2307.08307] [INSPIRE].
D. Patramanis and W. Sybesma, Krylov complexity in a natural basis for the Schrödinger algebra, arXiv:2306.03133 [INSPIRE].
A. Chattopadhyay, A. Mitra and H.J.R. van Zyl, Spread complexity as classical dilaton solutions, Phys. Rev. D 108 (2023) 025013 [arXiv:2302.10489] [INSPIRE].
N. Iizuka and M. Nishida, Krylov complexity in the IP matrix model, JHEP 11 (2023) 065 [arXiv:2306.04805] [INSPIRE].
K.-B. Huh, H.-S. Jeong and J.F. Pedraza, Spread complexity in saddle-dominated scrambling, arXiv:2312.12593 [INSPIRE].
K. Pal, K. Pal, A. Gill and T. Sarkar, Time evolution of spread complexity and statistics of work done in quantum quenches, Phys. Rev. B 108 (2023) 104311 [arXiv:2304.09636] [INSPIRE].
A. Bhattacharya, P. Nandy, P.P. Nath and H. Sahu, On Krylov complexity in open systems: an approach via bi-Lanczos algorithm, JHEP 12 (2023) 066 [arXiv:2303.04175] [INSPIRE].
T. Li and L.-H. Liu, Inflationary Krylov complexity, JHEP 04 (2024) 123 [arXiv:2401.09307] [INSPIRE].
T. Anegawa, N. Iizuka and M. Nishida, Krylov complexity as an order parameter for deconfinement phase transitions at large N, JHEP 04 (2024) 119 [arXiv:2401.04383] [INSPIRE].
C. Beetar et al., Complexity and Operator Growth for Quantum Systems in Dynamic Equilibrium, arXiv:2312.15790 [INSPIRE].
T.Q. Loc, Lanczos spectrum for random operator growth, arXiv:2402.07980 [INSPIRE].
K. Adhikari, S. Choudhury and A. Roy, Krylov complexity in quantum field theory, Nucl. Phys. B (2023) 116263 [https://doi.org/10.1016/j.nuclphysb.2023.116263].
K. Adhikari and S. Choudhury, Cosmological Krylov Complexity, Fortsch. Phys. 70 (2022) 2200126 [arXiv:2203.14330] [INSPIRE].
K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115 (2015) 211601 [arXiv:1502.06692] [INSPIRE].
A. del Campo, J. Molina-Vilaplana and J. Sonner, Scrambling the spectral form factor: unitarity constraints and exact results, Phys. Rev. D 95 (2017) 126008 [arXiv:1702.04350] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
Y. Chen, V. Ivo and J. Maldacena, Comments on the double cone wormhole, JHEP 04 (2024) 124 [arXiv:2310.11617] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
P. Caputa, J.M. Magan, D. Patramanis and E. Tonni, Krylov complexity of modular Hamiltonian evolution, Phys. Rev. D 109 (2024) 086004 [arXiv:2306.14732] [INSPIRE].
E. Caceres et al., Complexity of Mixed States in QFT and Holography, JHEP 03 (2020) 012 [arXiv:1909.10557] [INSPIRE].
R. Haag, Local Quantum Physics, Springer, Berlin (1996) [https://doi.org/10.1007/978-3-642-61458-3] [INSPIRE].
T. Ali et al., Chaos and Complexity in Quantum Mechanics, Phys. Rev. D 101 (2020) 026021 [arXiv:1905.13534] [INSPIRE].
L.-C. Qu, J. Chen and Y.-X. Liu, Chaos and complexity for inverted harmonic oscillators, Phys. Rev. D 105 (2022) 126015 [arXiv:2111.07351] [INSPIRE].
L.-C. Qu, H.-Y. Jiang and Y.-X. Liu, Chaos and multifold complexity for an inverted harmonic oscillator, JHEP 12 (2022) 065 [arXiv:2211.04317] [INSPIRE].
H. Tang, Operator Krylov complexity in random matrix theory, arXiv:2312.17416 [INSPIRE].
F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].
O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984) 1 [INSPIRE].
T. Guhr, A. Muller-Groeling and H.A. Weidenmuller, Random matrix theories in quantum physics: Common concepts, Phys. Rept. 299 (1998) 189 [cond-mat/9707301] [INSPIRE].
M.L. Mehta, On the statistical properties of the level-spacings in nuclear spectra, Nucl. Phys. 18 (1960) 395.
M. Gaudin, Sur la loi limite de l’espacement des valeurs propres d’une matrice aléatoire, Nucl. Phys. 25 (1961) 447.
F.J. Dyson, Statistical Theory of the Energy Levels of Complex Systems. III, J. Math. Phys. 3 (1962) 166 [INSPIRE].
B. Czech, Einstein Equations from Varying Complexity, Phys. Rev. Lett. 120 (2018) 031601 [arXiv:1706.00965] [INSPIRE].
L. Susskind, Complexity and Newton’s Laws, Front. in Phys. 8 (2020) 262 [arXiv:1904.12819] [INSPIRE].
J.F. Pedraza, A. Russo, A. Svesko and Z. Weller-Davies, Lorentzian Threads as Gatelines and Holographic Complexity, Phys. Rev. Lett. 127 (2021) 271602 [arXiv:2105.12735] [INSPIRE].
J.F. Pedraza, A. Russo, A. Svesko and Z. Weller-Davies, Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity, JHEP 02 (2022) 093 [arXiv:2106.12585] [INSPIRE].
J.F. Pedraza, A. Russo, A. Svesko and Z. Weller-Davies, Computing spacetime, Int. J. Mod. Phys. D 31 (2022) 2242010 [arXiv:2205.05705] [INSPIRE].
R. Carrasco, J.F. Pedraza, A. Svesko and Z. Weller-Davies, Gravitation from optimized computation: Einstein and beyond, JHEP 09 (2023) 167 [arXiv:2306.08503] [INSPIRE].
A. Kar, L. Lamprou, M. Rozali and J. Sully, Random matrix theory for complexity growth and black hole interiors, JHEP 01 (2022) 016 [arXiv:2106.02046] [INSPIRE].
Acknowledgments
We would like to thank José Barbón, Hugo A. Camargo, Bowen Chen, Kyoung-Bum Huh, Viktor Jahnke, Hong-Yue Jiang, Keun-Young Kim, Yu-Xiao Liu, Javier Magán, Mitsuhiro Nishida and Shan-Ming Ruan for valuable discussions and correspondence. PC and SL are supported by NAWA “Polish Returns 2019” PPN/PPO/2019/1/00010/U/0001 and NCN Sonata Bis 9 2019/34/E/ST2/00123 grants. PC would also like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Black Holes: bridges between number theory and holographic quantum information, where work on this paper was undertaken. HSJ and JFP are supported by the Spanish MINECO ‘Centro de Excelencia Severo Ochoa’ program under grant SEV-2012-0249, the Comunidad de Madrid ‘Atracción de Talento’ program (ATCAM) grant 2020-T1/TIC-20495, the Spanish Research Agency via grants CEX2020-001007-S and PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033, and ERDF A way of making Europe. LCQ acknowledges the financial support provided by the scholarship granted by the Chinese Scholarship Council (CSC). LCQ would also like to thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the completion of this work. All authors contributed equally to this paper and should be considered as co-first authors. LCQ is the corresponding author.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2402.09522
Rights and permissions
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.
About this article
Cite this article
Caputa, P., Jeong, HS., Liu, S. et al. Krylov complexity of density matrix operators. J. High Energ. Phys. 2024, 337 (2024). https://doi.org/10.1007/JHEP05(2024)337
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2024)337