Black Hole Complementarity and the Harlow-Hayden Conjecture

TL;DR

The paper assesses the AMPS firewall challenge to black hole complementarity by analyzing the entanglement structure of interior and exterior degrees of freedom and introducing a computational complexity perspective. It argues that the proposed proximity postulate leads to potential contradictions unless one accepts a nontrivial mapping between interior and exterior degrees of freedom, which can be reconciled if distilling fine-grained information is computationally intractable. The Harlow-Hayden conjecture is presented as a key mechanism: exponential-time distillation prevents Alice from violating causality and locality before evaporation, effectively protecting the no-firewall scenario. The discussion also emphasizes strong complementarity, where different observers may use frame-dependent, yet compatible, quantum descriptions. If correct, these ideas preserve black hole complementarity and reveal a deep connection between quantum information theory and gravitational physics.

Abstract

Black hole complementarity, as originally formulated in the 1990's by Preskill, 't Hooft, and myself is now being challenged by the Almheiri-Marolf-Polchinski-Sully firewall argument. The AMPS argument relies on an implicit assumption---the ``proximity postulate---which says that the interior of a black hole must be constructed from degrees of freedom that are physically near the black hole. The proximity postulate manifestly contradicts the idea that interior information is redundant with information in Hawking radiation, which is very far from the black hole. AMPS argue that a violation of the proximity postulate would lead to a contradiction in a thought-experiment in which Alice distills the Hawking radiation and brings a bit back to the black hole. According to AMPS the only way to protect against the contradiction is for a firewall to form at the Page time. But the measurement that Alice must make, is of such a fine-grained nature that carrying it out before the black hole evaporates may be impossible. Harlow and Hayden have found evidence that the limits of quantum computation do in fact prevent Alice from carrying out her experiment in less than exponential time. If their conjecture is correct then black hole complementarity may be alive and well. My aim here is to give an overview of the firewall argument, and its basis in the proximity postulate; as well as the counterargument based on computational complexity, as conjectured by Harlow and Hayden.

Figures (12)

• Figure 1: Dividing space into two entangled half-planes. The entanglement of a conformal theory can be envisioned in terms of mirror-image Bell pairs formed from cells on the right and left side.
• Figure 2: On the left side an $N$-qubit system is divided into a small subsystem with $M$ qubits and a big subsystem with $N-M$ qubits. On the right side the big subsystem is further divided into a second subsystem with $N$ qubits and a remainder of $N-2M$ qubits.
• Figure 3: The Page curve for distillable entanglement.
• Figure 4: The Black hole geometry is divided into four regions, ${\cal{R}}$, ${\cal{B}}$, ${\cal{H}}$, and ${\cal{A}}$. Region ${\cal{B}}$ is shown divided into thermal cells, each with a single bit of entropy.
• Figure 5: The pairing of $A$ and $B$ modes can be carried out on a space-like surface in an infalling frame.