Unitarity of black hole evaporation in final-state projection models

TL;DR

The paper investigates the Horowitz–Maldacena final-state projection as a mechanism to reconcile unitary black hole evaporation with horizon smoothness, by allowing postselected teleportation that effectively transfers information from collapsing matter to outgoing radiation. It shows that unitarity can be preserved for generic final states with exponentially small deviations $e^{-S_{BH}/2}$ and analyzes the conditions under which interior measurements might lead to acausal signaling, noting that decoding Hawking radiation could be computationally prohibitive. The work illustrates how relaxing no-cloning and monogamy of entanglement within a final-state framework can address the AMPS firewall puzzle, while also highlighting potential issues such as causality violations and the need for fine-tuning. Overall, it provides a provocative, if speculative, pathway toward understanding quantum gravity and the fate of information in black holes, while emphasizing substantial open questions about physical realizability and compatibility with holographic duals.

Abstract

Almheiri et al. have emphasized that otherwise reasonable beliefs about black hole evaporation are incompatible with the monogamy of quantum entanglement, a general property of quantum mechanics. We investigate the final-state projection model of black hole evaporation proposed by Horowitz and Maldacena, pointing out that this model admits cloning of quantum states and polygamous entanglement, allowing unitarity of the evaporation process to be reconciled with smoothness of the black hole event horizon. Though the model seems to require carefully tuned dynamics to ensure exact unitarity of the black hole S-matrix, for a generic final-state boundary condition the deviations from unitarity are exponentially small in the black hole entropy; furthermore observers inside black holes need not detect any deviations from standard quantum mechanics. Though measurements performed inside old black holes could potentially produce causality-violating phenomena, the computational complexity of decoding the Hawking radiation may render the causality violation unobservable. Final-state projection models illustrate how inviolable principles of standard quantum mechanics might be circumvented in a theory of quantum gravity.

Figures (16)

• Figure 1: The AMPS puzzle: An infalling observer, while still a safe distance from the singularity, is in causal contact with each of the systems $A$, $B$, and $R_B$. If $B$ were highly entangled with both $A$ and $R_B$, the observer should be able to verify the violation of entanglement monogamy.
• Figure 2: Quantum teleportation. To convey a quantum state $|\psi\rangle$ of system $A$ to system $C$, first a maximally entangled state $|\Phi(V)\rangle$ of $BC$ is prepared, and then $AB$ is projected to a maximally entangled state $|\Phi(U^*)\rangle$. To recover $|\psi\rangle$, a party at $C$ applies the unitary transformation $U^\dagger V^\dagger$.
• Figure 3: The Horowitz-Maldacena model, in which quantum information carried by the collapsing matter system $M$ is teleported out of a black hole. Outgoing Hawking radiation is maximally entangled with infalling radiation, and a final-state boundary condition projects $M$ and the infalling radiation to a maximally entangled state which encodes the unitary S-matrix $S$.
• Figure 4: Information flow for a black hole that maintains its mass by accreting a steady stream of infalling matter.
• Figure 5: (a) An old black hole is maximally entangled with system $R$ in the previously emitted Hawking radiation. After the black hole emits the Hawking quanta $B$, $R$ is entangled with $B$ and no longer entangled with the black hole. (b) Entanglement transfer in the HM model. $AB$ and $HR$ are maximally entangled Unruh vacuum states. The final-state projection of $HA$ onto a maximally entangled state creates maximal entanglement of $BR$ via entanglement swapping.