Relative Entanglement Entropies in 1+1-dimensional conformal field theories

Abstract

We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy $S({\rho }_1\Vert {\rho }_0)$ between two given reduced density matrices ${\rho }_1$ and ${\rho }_0$ of a quantum field theory, we employ the replica trick which relies on the path integral representation of $\mathrm{Tr}({\rho }_1{\rho }_0^{n-1})$ and define a set of Rényi relative entropies $S_n({\rho }_1\Vert {\rho }_0)$. We compute these quantities for integer values of the parameter $n$ and derive via the replica limit, the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator $i\partial \varphi$, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.

24 pages, 5 figures

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• Marginal quenches and drives in Tomonaga-Luttinger liquids
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• Entanglement in descendants