Hessian geometry and entanglement thermodynamics

TL;DR

This paper uses Hessian geometry to extend thermodynamics concepts to quantum entanglement, deriving entropy scaling laws and geometric representations that connect entanglement with spacetime geometry.

Contribution

It introduces a Hessian geometric framework for entanglement thermodynamics, linking entropy, geometry, and quantum critical systems in a novel way.

Findings

Derives entanglement entropy scaling for quantum critical systems.

Shows entanglement entropy can be represented as an integral of local information flow.

Identifies entangling surface with the domain boundary of the Hessian potential.

Abstract

We reconstruct entanglement thermodynamics by means of Hessian geometry, since this method exactly generalizes thermodynamics into much wider exponential family cases including quantum entanglement. Starting with the correct first law of entanglement thermodynamics, we derive that a proper choice of the Hessian potential leads to both of the entanglement entropy scaling for quantum critical systems and hyperbolic metric (or AdS space with imaginary time). We also derive geometric representation of the entanglement entropy in which the entropy is described as integration of local conserved current of information flowing across an entangling surface. We find that the entangling surface is equivalent to the domain boundary of the Hessian potential. This feature originates in a special property of critical systems in which we can identify the entanglement entropy with the Hessian potential after the second derivative by the canonical parameters, and this identification guarantees violation of extensive nature of the entropy.