# Canonical energy is quantum Fisher information

Journal of High Energy Physics, Apr 2016

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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP04%282016%29153.pdf

Nima Lashkari, Mark Van Raamsdonk. Canonical energy is quantum Fisher information, Journal of High Energy Physics, 2016, 153, DOI: 10.1007/JHEP04(2016)153