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 = 0t=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>0t>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.
Highlights
We will find that the complexity of formation of the thermofield double (TFD) state is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals, but that it does not exhibit the late-time growth characteristic of holographic theories/fast scramblers [73] due to the Gaussian nature of the TFD state for free scalar theories
We briefly review the approach of ref. [23] to circuit complexity, and demonstrate how the TFD state of two harmonic oscillators can be generated by quadratic operators
When studying the thermofield double state of an interacting field theory with a holographic dual, it is expected that circuit complexity can probe the degrees of freedom which are encoded in the dual geometry deep in the interior of black holes
Summary
In the context of holography [1], thermofield double (TFD) states play an especially important role. The left and right sides of the geometry associated with the black hole dual to the TFD state are connected by a wormhole, or Einstein-Rosen bridge (ERB), whose volume increases for a time which is exponential in the number of degrees of freedom of the boundary theory [12, 13] This time is much larger than other characteristic times in holography, e.g., the time for the mutual information to saturate or the scrambling time t∗. We will find that the complexity of formation of the TFD state is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals, but that it does not exhibit the late-time growth characteristic of holographic theories/fast scramblers [73] due to the Gaussian nature of the TFD state for free scalar theories We note that this problem has been studied previously in refs. We conclude with appendix G where we derive compact matrix functions for the time-dependent covariance matrix in position space, which we use for the efficient numerical evaluation of the entanglement entropy
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
