Abstract
We study upper bounds on the growth of operator entropy SK in operator growth. Using uncertainty relation, we first prove a dispersion bound on the growth rate |∂tSK| ≤ 2b1∆SK, where b1 is the first Lanczos coefficient and ∆SK is the variance of SK. However, for irreversible process, this bound generally turns out to be too loose at long times. We further find a tighter bound in the long time limit using a universal logarithmic relation between Krylov complexity and operator entropy. The new bound describes the long time behavior of operator entropy very well for physically interesting cases, such as chaotic systems and integrable models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman, A Universal Operator Growth Hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [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].
P. Caputa, J.M. Magan and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4 (2022) 013041 [arXiv:2109.03824] [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].
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].
A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104 (2021) L081702 [arXiv:2104.09514] [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].
D. Patramanis, Probing the entanglement of operator growth, PTEP 2022 (2022) 063A01 [arXiv:2111.03424] [INSPIRE].
K. Adhikari, S. Choudhury and A. Roy, \( \mathcal{K} \)rylov \( \mathcal{C} \)omplexity in \( \mathcal{Q} \)uantum \( \mathcal{F} \)ield \( \mathcal{T} \)heory, arXiv:2204.02250 [INSPIRE].
Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A 105 (2022) 062210 [arXiv:2202.07220] [INSPIRE].
N. Hörnedal, N. Carabba, A.S. Matsoukas-Roubeas and A. del Campo, Ultimate Physical Limits to the Growth of Operator Complexity, Commun. Phys. 5 (2022) 207 [arXiv:2202.05006] [INSPIRE].
V. Balasubramanian, P. Caputa, J. Magan and Q. Wu, Quantum chaos and the complexity of spread of states, arXiv:2202.06957 [INSPIRE].
V.S. Viswanath and G. Müller, The Recursion Method: Applications to Many-body Dynamics, Springer, Berlin, Germany (2008).
E.R. Davidson, On derivations of the uncertainty principle, J. Chem. Phys. 42 (1965) 1461.
M. Lee, Ergodic Theory, Infinite Products, and Long Time Behavior in Hermitian Models, Phys. Rev. Lett. 87 (2001) 250601.
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
M.H. Lee, J. Florencio and J. Hong, Dynamic equivalence of a two-dimensional quantum electron gas and a classical harmonic oscillator chain with an impurity mass, J. Phys. A 22 (1989) L331.
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: 2206.00855
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
Fan, ZY. The growth of operator entropy in operator growth. J. High Energ. Phys. 2022, 232 (2022). https://doi.org/10.1007/JHEP08(2022)232
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2022)232