Einstein's equations and the pseudo-entropy of pseudo-Riemannian information manifolds

TL;DR

This paper develops a geometric framework linking quantum information theory and gravity, deriving Einstein's equations from Fisher information metrics and exploring implications for cosmology and the nature of time.

Contribution

It introduces a novel derivation of Einstein's equations using statistical information geometry, revealing a quantum origin of the cosmological constant and a dynamical gravitational constant.

Findings

Derivation of Einstein's equations from Fisher pseudo-Riemannian metrics.

Identification of a quantum-origin positive cosmological constant.

Proposal of a non-Hermitian Hamiltonian linking non-unitary CFTs and information manifolds.

Abstract

Motivated by the corrected form of the entropy-area law, and with the help of von Neumann entropy of quantum matter, we construct an emergent spacetime by the virtue of the geometric language of statistical information manifolds. We discuss the link between Wald and Jacobson approaches of thermodynamic/gravity correspondence and Fisher pseudo-Riemannian metric of information manifold. We derive in detail Einstein's field equations in statistical information geometric forms. This results in finding a quantum origin of a positive cosmological constant that is founded on Fisher metric. This cosmological constant resembles those found in Lovelock's theories in a de Sitter background as a result of using the complex extension of spacetime and the Gaussian exponential families of probability distributions, and we find a time varying dynamical gravitational constant as a function of Fisher metric together with the corresponding Ryu-Takayanagi formula of such system. Consequently, we obtain a dynamical equation for the entropy in information manifold using Liouville-von Neumann equation from the Hamiltonian of the system. This Hamiltonian is suggested to be non-Hermitian, which corroborates the approaches that relate non-unitary conformal field theories to information manifolds. This provides some insights on resolving "the problem of time".