Fisher Information of a Black Hole Spacetime

Relativistic quantum metrology is the study of optimal measurement procedures within systems that have both quantum and relativistic components. Here we use Unruh-DeWitt detectors coupled to a massless scalar field as probes of thermal parameters in different spacetimes via a relativistic quantum metrology analysis. We consider both (2+1)-dimensional anti-de Sitter and BTZ black hole spacetimes. We compute the Fisher information to identify characteristics of the black hole spacetime and to compare it to a uniformly accelerating detector in anti-de Sitter space. We find the dependence of the Fisher information on temperature, detector energy gap, black hole mass, interaction time, and the initial state of the detector. We identify strategies that maximize the Fisher information and therefore the precision of estimation.