Canonical Energy is Quantum Fisher Information

TL;DR

This paper demonstrates that in holographic CFTs, the quantum Fisher information metric for vacuum perturbations is equivalent to the canonical energy in AdS spacetime, linking quantum information geometry to gravitational stability.

Contribution

It establishes a duality between quantum Fisher information and canonical energy, extending the understanding of perturbation constraints in holography and gravity.

Findings

Quantum Fisher information for a CFT region equals canonical energy in AdS.

Positivity of relative entropy implies positive definiteness of the Fisher information metric.

Second-order constraints extend linearized Einstein equations to nonlinear stability conditions.

Abstract

In quantum information theory, Fisher Information is a natural metric on the space of perturbations to a density matrix, defined by calculating the relative entropy with the unperturbed state at quadratic order in perturbations. In gravitational physics, Canonical Energy defines a natural metric on the space of perturbations to spacetimes with a Killing horizon. In this paper, we show that the Fisher information metric for perturbations to the vacuum density matrix of a ball-shaped region B in a holographic CFT is dual to the canonical energy metric for perturbations to a corresponding Rindler wedge R_B of Anti-de-Sitter space. Positivity of relative entropy at second order implies that the Fisher information metric is positive definite. Thus, for physical perturbations to anti-de-Sitter spacetime, the canonical energy associated to any Rindler wedge must be positive. This second-order constraint on the metric extends the first order result from relative entropy positivity that physical perturbations must satisfy the linearized Einstein's equations.