$(1+1)\mathrm{D}$ Calculation Provides Evidence that Quantum Entanglement Survives a Firewall

We analyze how preexisting entanglement between two Unruh-DeWitt particle detectors evolves when one of the detectors falls through a Rindler firewall in ($1+1$)-dimensional Minkowski space. The firewall effect is minor and does not wash out the detector-detector entanglement, in some regimes even preserving the entanglement better than Minkowski vacuum. The absence of cataclysmic events should continue to hold for young black hole firewalls. A firewall’s prospective ability to resolve the information paradox must hence hinge on its detailed gravitational structure, presently poorly understood.

Figure 1

Spacetime diagram of the two-detector system with the Rindler firewall. The dashed line at $t=x$ is the firewall. The solid lines are the worldlines of the Alice detector and the Bob detector, switched on at $t=0$ and off at $t=T>0$. Alice crosses the firewall during the detectors’ operation (at $t=T_f=x_A$ in the diagram) but Bob does not.

Figure 2

Negativity $\mathcal{N}$ of the Alice-Bob final state as a function of $x_A$ with the detector trajectories shown in Fig. 1 and with the sharp switching functions (17), for $R=4$, $T=1.8$, ${\mathrm{\Omega }}_A={\mathrm{\Omega }}_B=1$, and $\mathrm{\Lambda }={10}^{-2}\ll {\mathrm{\Omega }}_{\nu }$ and ${\lambda }_A={\lambda }_B=0.01$. Before the interaction $\mathcal{N}=\frac{1}{2}$, and the evolution causes the entanglement to degrade. When $x_A>T$, Alice does not cross the firewall during her detector’s operation and the entanglement degradation is identical to that in Minkowski vacuum. When $x_A<T$, Alice does cross the firewall, and the entanglement degradation depends nonmonotonically on $x_A$. Note that the leading order contribution to $\mathcal{N}$ is a homogeneous quadratic in (${\lambda }_A,{\lambda }_B$): an overall scaling of the two coupling constants (within the perturbative approximation) leaves the plot invariant up to a rescaling of the vertical axis.

Figure 3

As in Fig. 2 but with the Gaussian switching functions (18), with $\sigma =1$ and $t_0=2$, so that the Gaussian switching provides a smooth approximation to the sharp switching of Fig. 2. The firewall effect on Alice’s detector is now significant for $x_A\lesssim 3$. The qualitative properties are as in Fig. 2, including the nonmonotonicity in $x_A$.