Canonical energy is quantum Fisher information

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.