Entanglement at a Scale and Renormalization Monotones

We study the information content of the reduced density matrix of a region in quantum field theory that cannot be recovered from its subregion density matrices. We reconstruct the density matrix from its subregions using two approaches: scaling maps and recovery maps. The vacuum of a scale-invariant field theory is the fixed point of both transformations. We define the entanglement of scaling and the entanglement of recovery as measures of entanglement that are intrinsic to the continuum limit. Both measures increase monotonically under the renormalization group flow. This provides a unifying information-theoretic structure underlying the different approaches to the renormalization monotones in various dimensions. Our analysis applies to non-relativistic quantum field theories as well the relativistic ones, however, in relativistic case, the entanglement of scaling can diverge

Modular Hamiltonian of Excited States in Conformal Field Theory

We present a novel replica trick that computes the relative entropy of two arbitrary states in conformal field theory. Our replica trick is based on the analytic continuation of partition functions that break the replica Z_n symmetry. It provides a method for computing arbitrary matrix elements of the modular Hamiltonian corresponding to excited states in terms of correlation functions. We show that the quantum Fisher information in vacuum can be expressed in terms of two-point functions on the replica geometry. We perform sample calculations in two-dimensional conformal field theories.Comment: 5 pages, 1 figur

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.Comment: 26 pages, 1 figur

Universality of Quantum Information in Chaotic CFTs

We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of finite size in the infinite volume limit when the full system is an energy eigenstate. This reduced density matrix is close in trace distance to a density matrix, to which we refer as the ETH density matrix, that is independent of all the details of an eigenstate except its energy and charges under global symmetries. In two dimensions, the ETH density matrix is universal for all theories with the same value of central charge. We argue that the ETH density matrix is close in trace distance to the reduced density matrix of the (micro)canonical ensemble. We support the argument in higher dimensions by comparing the Von Neumann entropy of the ETH density matrix with the entropy of a black hole in holographic systems in the low temperature limit. Finally, we generalize our analysis to the coherent states with energy density that varies slowly in space, and show that locally such states are well described by the ETH density matrix.Comment: 43 page