Complexity and entanglement for thermofield double states

TL;DR

This work develops a covariant, Gaussian-state framework to compute circuit complexity for thermofield double states in free scalar QFTs using Nielsen geometry on the symplectic group. It shows that the complexity of formation is proportional to the thermodynamic entropy at t=0, while time evolution saturates at a time scale set by the inverse temperature, a feature tied to Gaussianity and absent in holographic models with chaotic dynamics. By comparing 1+1D entanglement dynamics with the complexity evolution, the authors reveal a nuanced picture: entanglement entropy can grow linearly and exhibit logarithmic zero-mode contributions, whereas complexity saturates due to limited access to non-Gaussian transformations. The work leverages a covariance-matrix formalism to efficiently track Bogoliubov transformations and provides explicit results across lattice and continuum QFT regularizations, clarifying both UV sensitivity and the role of the reference/gate scales. These findings offer a precise benchmark for holographic conjectures and raise open questions about how interactions and non-Gaussian gates might restore late-time complexity growth in more strongly coupled theories.

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.

Figures (30)

• Figure 1: Complexity=volume (CV, left) and complexity=action (CA, right) for the eternal AdS black hole dual to the TFD state \ref{['eq:TFD']}. Left panel: the blue curve represents the maximal spacelike surface that connects the specified time slices on the left and right boundaries. Right panel: the shaded region is the corresponding WdW patch. Figure and caption taken from ref. Jefferson:2017sdb.
• Figure 2: Single mode complexity as a function of time for $\beta \omega=0.5$ (blue), $1$ (yellow), and $2$ (green). We note that the complexity oscillates in time with periodicity $\delta t = \pi/\omega$ as expected from the explicit expressions and with an amplitude that decreases (approximately exponentially for large $\beta \omega$) for increasing $\beta\omega$.
• Figure 3: Illustration of the geometry of $\mathcal{F}_{r\to\textrm{\tiny T}}$ given in eq. \ref{['stasta3']} for different target states $|G_\textrm{\tiny T}\rangle$ and different values of $\lambda_\textrm{\tiny R}$. The black dot indicates the intersection with the $\tau=0$ plane, and all the curves originate from this same point.
• Figure 4: Complexity of a single mode TFD at $t=0$ with various values of $\lambda_\textrm{\tiny R}$ minus that with $\lambda_\textrm{\tiny R}=1$. In this plot we have fixed $\omega_\textrm{\tiny R}^2 \beta /M=10$ and $\beta \omega=1$ and this means that $\lambda_{\textrm{\tiny R}}/\lambda=10$ and focused on the $+$ mode. In this case we expect that for $1<\lambda_\textrm{\tiny R}<40.84$ we will have deviations from the straight line trajectories. Outside this range, we recover straight line trajectories and since the ratio of $\lambda/\lambda_\textrm{\tiny R}$ is fixed (($x-y$) is fixed), eq. \ref{['rhophi1']} predicts no dependence on $\lambda_\textrm{\tiny R}$ outside this range and this is indeed what we see in the figure when comparing to the $\lambda_\textrm{\tiny R}=1$ value. We remark that this plot does not have the resolution to show exponentially small deviations around $\Delta \mathcal{C}_{\kappa=2}=0$.
• Figure 5: Complexity of a single mode TFD with various values of $\lambda_\textrm{\tiny R}$ and of the parameters $\omega \beta$ and $\mu \beta$ as indicated on the different panels. The plot includes the sum of the $+$ and $-$ modes. The plot is always periodic as a function of $\omega t$ with periodicity $\pi$ as previously, and so we have only plotted one period. The result for the complexity is suppressed as we increase $\beta \omega$. When increasing the reference state scale encoded in the parameter $\mu \beta$, the curves for $\lambda_\textrm{\tiny R}<1$ approach the one of $\lambda_\textrm{\tiny R}=1$. We do not fully understand what is the reason for this behavior and leave it for future study. In some instances the (regularized) complexity is negative for all times.