  We examine geometry and dynamics of classical spacetime derived from
entanglement spectrum. The spacetime is a kind of canonical parameter space
defined by the Fisher information metric. As a concrete example, we focus on
the spectrum for free fermions in spatially one dimension. The spectrum has
exponential family form like thermal probability distribution owing to
mixed-state feature emerging from truncation of environmental degrees of
freedom. In this case, the Fisher metric is given by the second derivative of
the Hessian potential that can be identified with the entanglement entropy. We
emphasize that the canonical parameters are nontrivial functions of partial
system size by the truncation, filling fraction of fermions, and time. Then,
the precise determination of this nontrivial mapping is necessary to derive the
functional form of the Hessian potential that leads to correct entanglement
entropy scaling. By this potential, we find that the emergent geometry becomes
anti-de Sitter spacetime with imaginary time, and a radial axis as well as
spacetime coordinates appears spontaneously. We also find that the information
of the UV limit of the original free fermions lives in the boundary of the
anti-de Sitter spacetime. These findings strongly suggest that the Hessian
potential for free fermions has enough geometrical meaning associated with
gauge-gravity correspondence. Furthermore, some deformation of the spectrum
near the conformal fixed point is mapped onto spacetime dynamics. The
fluctuation of the entanglement entropy embedded into the spacetime behaves
like free scaler field, and the dynamics is described by the Einstein equation
with a negative cosmological constant. Therefore, the Einstein equation can be
regarded as the equation of original quantum state.
