## Complexity and entanglement for thermofield double states

Shira Chapman, Jens Eisert, Lucas Hackl, Michal P. Heller, Ro Jefferson, Hugo Marrochio, Robert C. Myers

SciPost Phys. 6, 034 (2019) · published 15 March 2019

- doi: 10.21468/SciPostPhys.6.3.034
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### 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.

### Ontology / Topics

See full Ontology or Topics database.### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1 }^{2 }Shira Chapman, -
^{3 }Jens Eisert, -
^{2 }^{4 }^{5 }Lucas Hackl, -
^{6 }Michal P. Heller, -
^{6 }Ro Jefferson, -
^{2 }^{7 }Hugo Marrochio, -
^{2 }Robert C. Myers

^{1}Institute of Physics, University of Amsterdam [IoP, UvA]^{2}Perimeter Institute [PI]^{3}Freie Universität Berlin / Freie Universität Berlin [FU Berlin]^{4}Pennsylvania State University [PSU]^{5}Max-Planck-Institut für Quantenoptik / Max Planck Institute of Quantum Optics [MPQ]^{6}Max-Planck-Institut für Gravitationsphysik / Max Planck Institute for Gravitational Physics [AEI]^{7}University of Waterloo [UW]

- Alexander von Humboldt-Stiftung / Alexander von Humboldt Foundation
- Bundesministerium für Bildung und Forschung / Federal Ministry of Education and Research [BMBF]
- Deutsche Forschungsgemeinschaft / German Research FoundationDeutsche Forschungsgemeinschaft [DFG]
- European Commission [EC]
- European Research Council [ERC]
- Gouvernement du Canada / Government of Canada
- Instituto Nazionale di Fisica Nucleare (INFN) (through Organization: Istituto Nazionale di Fisica Nucleare / National Institute for Nuclear Physics [INFN])
- John Templeton Foundation
- Ministry of Research and Innovation (Canada, Ontario, Min Res&Innov) (through Organization: Ministry of Research, Innovation and Science - Ontario [MRIS])
- National Science Foundation [NSF]
- Conseil de Recherches en Sciences Naturelles et en Génie / Natural Sciences and Engineering Research Council [NSERC / CRSNG]
- Simons Foundation