Multipartite entanglement and firewalls

Black holes offer an exciting area to explore the nature of quantum gravity. The classic work on Hawking radiation indicates that black holes should decay via quantum effects, but our ideas about how this might work at a technical level are incomplete. Recently Almheiri-Marolf-Polchinski-Sully (AMPS) have noted an apparent paradox in reconciling fundamental properties of quantum mechanics with standard beliefs about black holes. One way to resolve the paradox is to postulate the existence of a “firewall” inside the black hole horizon which prevents objects from falling smoothly toward the singularity. A fundamental limitation on the behavior of quantum entanglement known as “monogamy” plays a key role in the AMPS argument. Our goal is to study and apply many-body entanglement theory to consider the entanglement among different parts of Hawking radiation and black holes. Using the multipartite entanglement measure called negativity, we identify an example which could change the AMPS accounting of quantum entanglement and perhaps eliminate the need for a firewall. Specifically, we constructed a toy model for black hole decay which has different entanglement behavior than that assumed by AMPS. We discuss the additional steps that would be needed to bring lessons from our toy model to our understanding of realistic black holes.

Figure 1

For each time step we illustrate the allocation of our six qubits (labeled A–F) among the three physical roles: (i) part of the black hole, (ii) just radiated Hawking radiation, and (iii) other Hawking radiation. As time goes on the allocation shifts, corresponding to the decay process of the black hole.

Figure 2

Calculated von Neumann entropy of black hole subsystem (x’s) compared to the maximum possible value of von Neumann entropy in our multipartite model (+’s) at each time step. The maximum values of entropy are based on the dimension of the smaller Hilbert space between the measured subsystem and the rest of the space which together form a purification. This plot exhibits standard behaviors discussed by Page [21].