# Complexity and entanglement for thermofield double states

### Submission summary

 As Contributors: Shira Chapman · Lucas Hackl · Michal P. Heller · Ro Jefferson Arxiv Link: https://arxiv.org/abs/1810.05151v3 (pdf) Date submitted: 2019-02-11 01:00 Submitted by: Hackl, Lucas Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Quantum Physics Approach: Theoretical

### 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.

###### Current status:
Has been resubmitted

Dear Editor,

We would like to thank the referees for their careful reading and the detailed reports. We replied to all points raised by the referees in our direct replies. We believe that the updated version has improved clarify and we hope that it now accepted for publication in SciPost.

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

### List of changes

Based on the three referee reports, we implemented the following changes (for more details see our reply to the referees):
- We corrected the typo (or -> for) pointed out.
- We adjusted figure 7, where we now clearly refer to individual curves.
- We included a definition for $t_R$ and $t_L$ after eq. (1).
- We clarified refs. [74-75] and also added 1608.00614.
- We added a clarification regarding the vertical axis of figure 4.
- We corrected a typo (n -> k) in eq. (196) and (197).
- We added a comment regarding the early linear growth of $\Delta\mathcal{C}_1^{\mathrm{LR}}$ below eq. (160).
- We added a clarification (new footnote 42) regarding the early growth shown in figure 20.
- Following the referee's suggestion, we weakened our initial claim on what we can say analytically about the zero more, which now states: “In this subsection, we analyze the time dependence of the entanglement entropy for a single degree of freedom in the limit m → 0 and find a logarithmic contribution. We then argue that the extracted asymptotic behavior is due to the zero mode and also applies to larger subsystem."
- We elaborated on the relationship between the Lieb-Robinson bound and the quasi-particle picture, which can be found on page 63.

### Submission & Refereeing History

Resubmission 1810.05151v4 on 18 February 2019