We examine the Bañados–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 CFT1+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.
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