Gravity dualities 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. the trace distance, Fubini-Study distance, Bures distance and R\'{e}nyi relative entropy, all have gravity dualities. Their gravity dualities have two equivalent descriptions: one is given by the integration of the area of a cosmic brane, the other one is given by the Euclidian on-shell action of dual theory and the area of the cosmic brane. It then applies these dualities into the 2-dimensional conformal field theory as examples and finds the results match with the computations of field theory exactly.