Correspondence between causality in flat Minkowski spacetime and entanglement in thermofield-double state: Hessian-geometrical study

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.