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Complexity and entanglement for thermofield double states

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

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Shira Chapman · Lucas Hackl · Michal P. Heller · Ro Jefferson
Submission information
Preprint Link:  (pdf)
Date accepted: 2019-02-25
Date submitted: 2019-02-18 01:00
Submitted by: Hackl, Lucas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


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 comments upon resubmission

We apologize for missing the reference 'P. Calabrese and J. Cardy, J. Stat. Mech. P04010 (2005)'. This work should indeed be cited before Ref. [92] and [93] when the introduction of the quasi-particle picture is discussed. Again, we apologize for the omission.

List of changes

We added 'P. Calabrese and J. Cardy, J. Stat. Mech. P04010 (2005)' as new reference [92].

Published as SciPost Phys. 6, 034 (2019)

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