  We examine the Hessian potential that derives the flat Minkowski spacetime in
$(1+1)$-dimension. The entanglement thermodynamics by the Hessian geometry
enables us to obtain the entanglement entropy of a corresponding quantum state
by means of holography. We find that the positivity of the entropy leads to the
presence of past and future causal cones in the Minkowski spacetime. We also
find that the quantum state is equivalent to the thermofield-double state, and
then the entropy is proportional to the temperature. The proportionality is
consistent with previous holographic works. The present Hessian geometrical
approach captures that the causality in the classical side is converted into
quantum entanglement inherent in the thermofield dynamics.
