Relative entropy formulation of thermalization process in a Schwarzschild spacetime

We revisit the problem of the thermalization process in an entropic formulation for the Unruh-DeWitt (UDW) detector outside a Schwarzschild black hole. We derive the late-time dynamics of the detector in the context of open quantum system, and capture the path distinguishability and thermodynamic irreversibility of detector thermalization process by using quantum relative entropy (QRE). We find that beyond the Planckian transition rate, the refined thermalization process in detector Hilbert space can be distinguished by the time behavior of the related QRE. We show that the exotic position-dependent behaviors of the QRE emerge corresponding to different choices of black hole vacua (i.e., the Boulware, Hartle-Hawking, and Unruh vacua). Finally, from a perspective of quantum thermodynamics, we recast the free energy change of the UDW detector undergoing Hawking radiation into an entropic combination form, where the classical Kullback-Leibler divergence and quantum coherence are presented in specific QRE-like forms. With growing Hawking temperature, we find that the consumption rate of quantum coherence is larger than that of its classical counterpart.