@article{97dedba686a1418daeed700b4db3e167,
title = "Complexity and Entanglement for Thermofield Double States",
abstract = "Motivated by holographic complexity proposals as novel probes of black hole spacetimes, we explore circuit complexity for thermofield double (TFD) states in free scalar quantum field theories using the Nielsen approach. For TFD states at t = 0, we show that the complexity of formation is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals. For TFD states at t > 0, we demonstrate that the complexity evolves in time and saturates after a time of the order of the inverse temperature. The latter feature, which is in contrast with the results of holographic proposals, is due to the Gaussian nature of the TFD state of the free bosonic QFT. A novel technical aspect of our work is framing complexity calculations in the language of covariance matrices and the associated symplectic transformations, which provide a natural language for dealing with Gaussian states. Furthermore, for free QFTs in 1+1 dimension, we compare the dynamics of circuit complexity with the time dependence of the entanglement entropy for simple bipartitions of TFDs. We relate our results for the entanglement entropy to previous studies on non-equilibrium entanglement evolution following quenches. We also present a new analytic derivation of a logarithmic contribution due to the zero momentum mode in the limit of vanishing mass for a subsystem containing a single degree of freedom on each side of the TFD and argue why a similar logarithmic growth should be present for larger subsystems.",
author = "Shira Chapman and Jens Eisert and Lucas Hackl and Heller, {Michal P.} and Ro Jefferson and Hugo Marrochio and Myers, {Robert C.}",
note = "Funding Information: We would like to thank J. de Boer, P. Calabrese, P. Caputa, J. Hernandez, K. Papadodimas, S. M. Ruan, M. Smolkin, T. Takayanagi, E. Verlinde, M. Walter, and the members of the QGFI group at AEI for useful comments and discussions. We would also like to extend special thanks to F. Pastawski for collaboration during the initial stages of this project. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science. JE, MPH, HM, and RCM thank the Kavli Institute for Theoretical Physics for its hospitality at various stages of this project. At the KITP, this research was supported in part by the National Science Foundation under Grant No. NSF PHY17-48958. SC and RCM also thank the Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial support where part of this work was carried out. MPH would also like to thank University of Amsterdam, Chulalongkorn University, Hanyang University, Hebrew University of Jerusalem, Mainz Institute for Theoretical Physics, National Taiwan University, University of Warsaw and Yukawa Institute for Theoretical Physics for extended hospitality during the completion of this project. LH thanks the Visiting Graduate Fellow program for hospitality at the Perimeter Institute during this project. LH was also supported by a Frymoyer fellowship, a Mebus fellowship, by the National Science Foundation under Grant No. PHY-1404204 awarded to Eugenio Bianchi and the Max Planck Harvard Research Center for Quantum Optics. RCM is supported by funding from the Natural Sciences and Engineering Research Council of Canada and from the Simons Foundation through the “It from Qubit” collaboration. MPH and RJ acknowledge support from the Alexander von Humboldt Foundation and the Federal Ministry for Education and Research through the Sofja Kovalevskaja Award. JE is supported by the DFG (EI 519/14-1, EI 519/7-1, CRC 183, FOR 2724), the Templeton Foundation, and the ERC (TAQ). This work has also received funding from the European Union{\textquoteright}s Horizon 2020 research and innovation programme under grant agreement No 817482 (PASQUANS). SC acknowledges funding from the European Research Council (ERC) under starting grant No. 715656 (GenGeoHol) awarded to Diego M. Hofman. Publisher Copyright: Copyright S. Chapman et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation.",
year = "2019",
month = mar,
day = "15",
doi = "10.21468/SciPostPhys.6.3.034",
language = "English",
volume = "6",
pages = "34",
journal = "SciPost Physics",
issn = "2542-4653",
publisher = "SciPost Foundation",
number = "3",
}