The Transfer of Entanglement: The Case for Firewalls

TL;DR

Black hole complementarity ties interior geometry to exterior degrees of freedom via entanglement; as a black hole evaporates, entanglement shifts from near-horizon modes to Hawking radiation, and by the Page time the interior structure is largely depleted, suggesting a firewall replaces the horizon in typical states. The authors develop a qubit-based model with zone ${\cal Z}$ and stretched horizon ${\cal H}$, and a localized pull-back---push-forward procedure to map interior operators to exterior degrees of freedom, providing quantitative bounds via mutual information and distillable entanglement. They show how entanglement degenerates during evaporation, constrain the interior DOF by the available ${\cal Z}$–${\cal H}$ entanglement, and discuss radical implications for horizon smoothness and potential alternatives. The work highlights deep tensions between unitarity, locality, and the experience of infalling observers, and frames firewalls as a natural outcome for typical black-hole states at or after Page time.

Abstract

Black hole complementarity requires that the interior of a black hole be represented by the same degrees of freedom that describe its exterior. Entanglement plays a crucial role in the reconstruction of the interior degrees of freedom. This connection is manifest in "two-sided" eternal black holes. But for real black holes which are formed from collapse there are no second sides. The sense in which horizon entropy is entanglement entropy is much more subtle for one-sided black holes. It involves entanglement between different parts of the near-horizon system. As a one-sided black hole evaporates the entanglement that accounts for its interior degrees of freedom disappears, and is gradually replaced by entanglement with the outgoing Hawking radiation. A principle of "transfer of entanglement" can be formulated. According to the argument of Almheiri, Marolf, Polchinski and Sully, it is when the transfer of entanglement is completed at the Page time, that a firewall replaces the horizon. Alternatives to firewalls may suffer contradictions which are similar to those of time travel. The firewall hypothesis would be similar to Hawking's chronology protection conjecture.

Figures (7)

• Figure 1: The pull-back---push-forward strategy. The field operator is pulled back to the remote past using the equations of motion in the causal past of $A$ shown as pink. Then it is pushed forward using the exterior equations of motion. On the left the operator is pushed all the way to the infinite future in order to express it in terms of Hawking radiation. On the right the operator is pushed forward to a finite time-slice where it is expressed in terms of the exterior degrees of freedom.
• Figure 2: A Schwarzschild time can be identified with every point behind the horizon. The grey curve represents the stretched horizon about a Planck length outside the horizon. The modes $A$ and $B$ are on opposite sides of the horizon at the same time.
• Figure 3: Foliating the geometry by constant $t$ slices. Outside the stretched horizon the slices are defined by Schwarzschild time. Inside the horizon they are light-sheets.
• Figure 4: Penrose Diagram for an ADS black hole. The diagram is composed of four quadrants. Quadrant I and II represent the physical black hole.
• Figure 5: The mode at $B$ is entangled with a mode at $B'$. Both $B$ and $B'$ are outside their respective horizons. It is clear from the diagram that the mode $A,$ which is situated near $B,$ but behind the horizon, is the same as $B'.$ The strong entanglement between $A$ and $B$ is nothing but the entanglement between $B$ and $B'.$