  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.
