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.
Highlights
But rather two complementary views of the same system, related by a complex nonlocal map
Almheiri et al (AMPS) recently argued [8] that these three assumptions are incompatible. They consider the Hawking radiation B emitted by a black hole which is nearly maximally entangled with an exterior system R. (For example, R could be the radiation so far emitted by an old black hole which has already radiated away more than half of its initial entropy [9, 10].) Assumptions (1) and (3) require B to be highly entangled with a subsystem RB of R, while assumption (2) requires B to be highly entangled with a subsystem A in the black hole interior
AMPS investigated the compatibility of three reasonable assumptions: (1) unitarity of black hole evaporation, (2) smoothness of the black hole event horizon, and (3) validity of local effective field theory outside a black hole
Summary
The HM proposal is based on quantum teleportation [35], which is illustrated in figure 2. Because the final-state boundary condition specifies that only one particular maximally entangled state is accepted at the singularity, there is no need for classical communication to convey the outcome of the entangled measurement. (The microcanonical ensemble is appropriate if we wish to consider the formation and evaporation of a black hole with sharply defined energy; an observer with access to a small subsystem of Hout will see a thermal state.) By a suitable basis choice, we set the unitary matrix specifying this maximally entangled state to the identity. Because the final-state boundary condition accepts any quantum state of the infalling matter system, observers approaching the singularity, those with access to only a local subsystem, need not experience a reversal in the arrow of time or any departure from the usual laws of quantum mechanics.
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