  We examine the Banados-Teitelboim-Zanelli (BTZ) black hole in terms of the
information geometry and consider what kind of quantum information produces the
black hole metric in close connection with the anti-de Sitter space/conformal
field theory (AdS/CFT) correspondence. We find a Hessian potential that exactly
produces both the BTZ metric and the entanglement entropy formula for CFT_{1+1}
at a finite temperature. Taking a free-falling frame near the event horizon is
a key procedure to derive these exact results. We also find an alternative
Hessian potential that produces the same BTZ metric, which is found using the
duality relation based on the Legendre transformation. We realize that the dual
representation originates from the entanglement Hamiltonian on the CFT side.
Our results suggest that the present information-geometrical approach is very
powerful for understanding the mechanism of the holographic renormalization
group such as the AdS/CFT correspondence.
