Horizons as Eavesdroppers: Horizon Algebras and Soft Quantum Information

It has recently been shown that a black hole or cosmological horizon will eventually decohere any quantum superposition in its vicinity. This occurs because, in effect, the long-range fields sourced by the superposition register on the horizon which forces the emission of entangling "soft gravitons/photons" through the horizon. We sharpen and generalize this mechanism by recasting the original gedankenexperiment in the language of (approximate) quantum error correction. This yields a complementary picture where the decoherence is due to an "eavesdropper" (Bob) inside the black hole attempting to obtain "which path" information by measuring the long-range fields of the superposed body. To quantify this we construct a generalized algebra of observables for which infrared-finite states are defined, and provide an infrared extension of Tomita-Takesaki theory for the resulting "horizon algebra." We compute the quantum fidelity to determine the amount of information such an interior observer can obtain, and derive a direct relationship between Bob's information and the decoherence of the superposition in the exterior. In particular we show that the decoherence of the superposition corresponds to the distinguishability of outcomes of the "optimal" measurement performable in the black hole interior.