Entanglement entropy of disjoint regions in excited states: an operator method

Abstract

We develop the computational method of entanglement entropy based on the idea that $$ \mathrm{T}\mathrm{r}{\rho}_{{}^{\varOmega}}^n $$ is written as the expectation value of the local operator, where ρ Ω is a density matrix of the subsystem Ω. We apply it to consider the mutual Rényi information I (n)(A, B) = S () + S () − S ∪ () of disjoint compact spatial regions A and B in the locally excited states defined by acting the local operators at A and B on the vacuum of a (d + 1)-dimensional field theory, in the limit when the separation r between A and B is much greater than their sizes R A,B . For the general QFT which has a mass gap, we compute I (n)(A, B) explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. For a free massless scalar field, we show that for some classes of excited states, I (n)(A, B) − I (n)(A, B)| r → ∞ = C () /r α(d − 1) where α = 1 or 2 which is determined by the property of the local operators under the transformation ϕ → −ϕ and α = 2 for the vacuum state. We give a method to compute C (2) systematically.