Canonical energy is quantum Fisher information
Author(s)Lashkari, Nima; Van Raamsdonk, Mark
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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[subscript 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.
DepartmentMassachusetts Institute of Technology. Center for Theoretical Physics; Massachusetts Institute of Technology. Laboratory for Nuclear Science
Journal of High Energy Physics
Springer Berlin Heidelberg
Lashkari, Nima, and Mark Van Raamsdonk. "Canonical energy is quantum Fisher information." Journal of High Energy Physics (April 2016), 2016:153, pp.1-25.
Final published version