Non-Markovian Gravitational Decoherence and the Black Hole Information Paradox

An isolated system always evolves according to unitary evolution and maintain coherence. However the system inevitably interacts with the environment, information about the relative phases between the quantum states leaks into the environment and becomes delocalized (known as environment-induced decoherence). Here we consider the gravitational decoherence of a quantum system near the event horizon of a Schwarzschild black hole. We show that the gravitational decoherence is non-Markovian, which is consistent with information conservation and the unitarity of quantum mechanics. Moreover, taking the point of view that information on the collapsed matter is stored as the quantum fluctuation of the horizon, the horizon can be regarded as an "uncertain" quantum object instead of random fluctuations of gravitons. Now the Hawking radiation bath plays the role of the environment, leading to the decoherence of the horizon. Therefore, information about the location of the horizon, or equivalently information about the collapsed matter, is carried away by the Hawking radiation. We also investigate the time dependence of the entanglement entropy of the Hawking radiation.