2018/10/31 by Shira Chapman, Jens Eisert, Lucas Hackl +4 · 2 citations
Physics and Astronomy · #Arrow of time #Black Holes and Theoretical Physics #Circuit complexity #Entropy (arrow of time) #Gaussian #Holography #Logarithm #Noncommutative and Quantum Gravity Theories #Quantum #Quantum Electrodynamics and Casimir Effect #Quantum entanglement #cond-mat.stat-mech #hep-th #quant-ph
paper · pdf · doi:10.21468/scipostphys.6.3.034
published as SciPost Phys. 6, 034 (2019) · 85+30 pages, 30 figures, 7 appendices; v4: reference added;
openalex created_date 2018/10/26 · arxiv created 2019/02/15 · openalex publication_date 2019/03/15 · arxiv updated 2019/03/18 · openalex updated_date 2026/08/06
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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.