Gravity duals of quantum distances

By choosing modular ground state as the reference state, this paper finds that three most frequently-used distances and a quantum quasi-distance, i.e. trace distance, Fubini-Study Bures distance R\'{e}nyi relative entropy, all have gravity dualities. Their dualities two equivalent descriptions: one is given by integration of area cosmic brane, other Euclidian on-shell action dual theory brane. It then applies these into 2-dimensional conformal field examples results match with computations exactly.