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Canonical Energy is Quantum Fisher Information

Nima Lashkari, Mark Van Raamsdonk

TL;DR

This work establishes a precise duality between quantum Fisher information for perturbations of boundary ball regions in a holographic CFT and the canonical energy of corresponding bulk Rindler wedges in AdS. By leveraging the Hollands–Wald identity, the authors show that second-order positivity of relative entropy in the CFT translates into a positive canonical energy condition in the bulk, thereby extending linearized Einstein constraints to second order. The framework yields an explicit bulk energy constraint ${oldsymbol E}_B(\delta g, \\delta g) \,= \, {oldsymbol E}^{ ext{grav}}_B(\\delta g, \\delta g) - 2 \, \int_\Sigma \, \xi^a T^{(2)}_{ab} \, \epsilon^b \ge 0$, linking geometry and matter in a covariant way. Concrete AdS$_3$ examples demonstrate how to compute canonical energy from a given perturbation and confirm consistency with the second-order relative-entropy calculation. These results deepen the connection between quantum information metrics and spacetime energy conditions, offering a path toward quantum-gravitational insights from entanglement structure.

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.

Canonical Energy is Quantum Fisher Information

TL;DR

This work establishes a precise duality between quantum Fisher information for perturbations of boundary ball regions in a holographic CFT and the canonical energy of corresponding bulk Rindler wedges in AdS. By leveraging the Hollands–Wald identity, the authors show that second-order positivity of relative entropy in the CFT translates into a positive canonical energy condition in the bulk, thereby extending linearized Einstein constraints to second order. The framework yields an explicit bulk energy constraint , linking geometry and matter in a covariant way. Concrete AdS examples demonstrate how to compute canonical energy from a given perturbation and confirm consistency with the second-order relative-entropy calculation. These results deepen the connection between quantum information metrics and spacetime energy conditions, offering a path toward quantum-gravitational insights from entanglement structure.

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.

Paper Structure

This paper contains 16 sections, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Relative Entropy and Fisher Information
  4. Relative entropy in conformal field theories
  5. Relative entropy in holographic conformal field theories
  6. Constraints on spacetime geometry from relative entropy inequalities
  7. A fundamental identity
  8. Bulk integral for relative entropy
  9. First order results
  10. Second order results: the gravity dual of Fisher Information
  11. Gravitational constraints from positivity of Fisher Information
  12. Examples
  13. Transformation to Hollands-Wald gauge
  14. Calculating canonical energy from $h$ and $V$
  15. Example: perturbations to Poincare $AdS_3$
  16. ...and 1 more sections

Figures (1)

  • Figure 1: AdS-Rindler wedge $R_B$ associated with a ball $B$ on a spatial slice of the boundary. $R_B$ is the intersection of the causal past and the causal future of the domain of dependence $D_B$ (boundary diamond). Solid blue paths indicate the boundary flow associated with $H_B$ and the conformal Killing vector $\zeta$. Dashed red paths indicate the action of the Killing vector $\xi$.