Quantum information metric of conical defect

A concept of measuring the quantum distance between two different quantum states which is called quantum information metric is presented. The holographic principle (AdS/CFT) suggests that the quantum information metric $G_{\lambda\lambda}$ between perturbed state and unperturbed state in field theory has a dual description in the classical gravity. In this work we calculate the quantum information metric of a theory which is dual to a conical defect geometry and we show that it is $n$ times the one of its covering space. We also give a holographic check for our result in the gravity side. Meanwhile, it was argued that $G_{\lambda\lambda}$ is dual to a codimension-one surface in spacetime and satisfies $G_{\lambda\lambda}=n_{d}\cdot\mbox{Vol}(\Sigma_{max})/L^{d}$. We show that the coefficient $n_d$ for conical defect should be rescaled by $n^2$ from the one for AdS. A limit case of conical defect --- the massless BTZ black hole--- is also considered. We show that the quantum information metric of the massless BTZ black hole disagrees with the one obtained by taking the vanishing temperature limit in BTZ black hole. This provides a new arena in differiating the different phases between BTZ spacetime and its massless cousin.