Notes on Quantum Entanglement of Local Operators

TL;DR

This work rigorously analyzes Renyi entanglement entropies for excited states generated by local operators in a free massless scalar field theory using the replica method. It demonstrates that the final excess entropies for operators built from a single-species field component follow a binomial-distribution structure, and that multi-operator excitations satisfy additive sum rules. The time evolution of these entropies maps to relativistic propagation of entangled pairs, with two-dimensional cases providing clear EPR-like saturation and higher dimensions yielding dimension-independent late-time limits. A unified propagator-based argument corroborates the binomial form and sum rules, and the results are extended to S-operators (single-species composites) with generalizations to derivatives and exponentials, while addressing conical singularity effects and outlining open directions for interacting or massive theories.

Abstract

This is an expanded version of the short report arXiv:1401.0539, where we stud- ied the (Renyi) entanglement entropies for the excited state defined by acting a given local operator on the ground state. We introduced the (Renyi) entanglement entropies of given local operators which measure the degrees of freedom of local operators and characterize them in conformal field theories from the viewpoint of quantum entanglement. In present paper, we explain how to compute them in free massless scalar field theories and we also investigate their time evolution. The results are interpreted in terms of relativistic propagation of an entangled pair. The main new results which we acquire in the present paper are as follows. Firstly, we provide an explanation which shows that the (Renyi) entanglement entropies of a specific operator are given by (Renyi) entanglement entropies of binomial distribution by the replica method. That operator is constructed of only scalar field. Secondly, we found the sum rule which (Renyi) entanglement entropies of those local operators obey. Those local operators are located separately. Moreover we argue that (Renyi) entanglement entropies of specific operators in conformal field theories are given by (Renyi) entanglement entropies of binomial distribution. These specific operators are constructed of single-species operator. We also argue that general operators obey the sum rule which we mentioned above.

Figures (14)

• Figure 1: After analytic continuation, $r_1, r_2, \theta_1,$ and $\theta_2$ are related to $\epsilon, t_1$ and $l$ as in this figure.
• Figure 2: $n$-sheeted geometry $\Sigma_n$ is constructed by gluing subsystem A on a sheet to subsystem A on the next sheet. Fields acquire the phase shift of $2\pi$ if they go around the boundary of A on the each sheet of $\Sigma_n$.
• Figure 3: The subsystem A is given by the region $x_1 \ge 0, \tau=0$. Operators are located at $(r, \theta, {\bf x}), (s, \theta' , {\bf x}')$ respectively. ${\bf x}$ are the coordinates along the direction vertical to the plane which has a conical singularity.
• Figure 4: The purple line corresponds to the integral contour in the complex plane.
• Figure 5: The plots of $\Delta S^{(2)}_A$ as functions of $t_1$ in the $\epsilon \rightarrow 0$ limit. The vertical line corresponds to $\Delta S^{(2)}_A$. And the horizontal line corresponds to $t_1$. Here we chose $l=10$. The red curve corresponds to the plot of $\Delta S^{(2)}_A$ for $|c| = \frac{1}{2}$. The blue curve corresponds to the plot of $\Delta S^{(2)}_A$ for $|c| = 1$.