Holographic evolution of the mutual information

TL;DR

This paper analyzes the time evolution of entanglement measures in holographic quenches described by Vaidya spacetimes in AdS_3 and AdS_4. By computing time-dependent holographic entanglement entropy for single regions and mutual information for disjoint regions, it reveals non-monotonic MI dynamics, transitions between connected and disconnected bulk surfaces, and a region where MI vanishes at all times. The authors show that the null energy condition is necessary for both strong subadditivity and monogamy of holographic mutual information, and they demonstrate how NEC-violating mass profiles can spoil these inequalities and the associated monogamy relations via tripartite information I3. The results bridge geometric energy conditions with information-theoretic inequalities in dynamical holographic settings and sharpen our understanding of entanglement structure during holographic thermalization.

Abstract

We compute the time evolution of the mutual information in out of equilibrium quantum systems whose gravity duals are Vaidya spacetimes in three and four dimensions, which describe the formation of a black hole through the collapse of null dust. We find the holographic mutual information to be non monotonic in time and always monogamous in the ranges explored. We also find that there is a region in the configuration space where it vanishes at all times. We show that the null energy condition is a necessary condition for both the strong subadditivity of the holographic entanglement entropy and the monogamy of the holographic mutual information.

Figures (16)

• Figure 1: The function (\ref{['mass pos kink']}) for different values of the thickness $a_v$ and $M=1$. The dashed curve is a step function and corresponds to the the limiting regime of thin shell $a_v \rightarrow 0$.
• Figure 2: Finite term of the holographic entanglement entropy for the Vaidya metric in three dimensions (see (\ref{['RT formula']}) and (\ref{['Ad reg def']}) up to a factor) with $m(v)$ given by (\ref{['mass pos kink']}) Hubeny:2007xtAbajoArrastia:2010yt. On the left, $L_{2,\textrm{reg}}(\ell,t)$ for different values${^\dagger}$ of boundary time $t$, increasing from the red curve to the blue one, at a fixed value of $a_v$. On the right, $L_{2,\textrm{reg}}(\ell,a_v)$ at a fixed value of $t$ and different values of the thickness $a_v$, which increases going from the blue curve to the red one. In both the plots, the black curves correspond to the limiting regimes of $AdS_3$ (bottom curve, from (\ref{['Lreg ads3 btz']})) and of the BTZ blak hole (top curve, from (\ref{['Lreg ads3 btz']})), while the dashed curves represent the corresponding curves for the thin shell limit $a_v=0$.
• Figure 3: Finite term of the holographic entanglement entropy for the Vaidya metric in four dimensions (proportional to $L_{3,\textrm{reg}}(\ell,t)$) with $m(v)$ given by (\ref{['mass pos kink']}), fixed $a_v$ and different boundary times $t$, increasing from the red curve to the blue one. The black curves correspond to the limiting regimes of $AdS_4$ (bottom curve from (\ref{['Lreg ads']})) and of the Schwarzschild black hole in four dimensions (top curve).
• Figure 4: Geodesics configuration for the holographic mutual information at the transition point for three dimensional Vaidya geometry ($d=2$) with $M=1$ in (\ref{['mass pos kink']}). The total length of the geodesics for the connected configuration (red) and the disconnected one (blue) is the same. In the upper plots $a_v = 0.5$ while in the bottom ones $a_v \rightarrow 0$ (thin shell limit). The boundary time is $t = 7$ (see the intersection of the curves with the horizontal axis in the plots on the right). The $z$ axis has been compactified using the $\arctan$ function. The green geodesics represent the mixed configuration, which is suboptimal. Notice that they do not intersectas it can be clearly seen from the plot on the right, top line.
• Figure 5: Holographic mutual information $I(\ell_0, \ell_1, \ell_1)$ in terms of $\ell_0$ for Vaidya metrics in three (plots on the left, infinitely thin shell regime) and four dimensions (plots on the right) in the bulk. Different curves are characterized by the boundary time $t$, whose value increases going from the red curves to the blue ones with $\Delta t =1$ and within the range indicated. The black curves correspond to $AdS_{d+1}$ (top curve) and Schwarzschild black hole (bottom curve). The transition of the holographic mutual information is continuos with a discontinuous first derivative.