Quantum Entanglement of Local Operators in Conformal Field Theories

TL;DR

A series of quantities which characterize a given local operator in any conformal field theory from the viewpoint of quantum entanglement are introduced, defined by the increased amount of (Rényi)Entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum.

Abstract

We introduce a series of quantities which characterizes a given local operator in conformal field theories from the viewpoint of quantum entanglement. It is defined by the increased amount of (Renyi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum. We consider a conformal field theory on an infinite space and take the subsystem in the definition of the entanglement entropy to be its half. We calculate these quantities for a free massless scalar field theory in 2, 4 and 6 dimensions. We find that these results are interpreted in terms of quantum entanglement of finite number states, including EPR states. They agree with a heuristic picture of propagations of entangled particles.

Figures (5)

• Figure 1: The $n$-sheeted geometry $\Sigma_n$ is constructed by gluing the upper cut along subsystem A on a sheet to the lower cut on the next sheet.
• Figure 2: The Euclidean coordinate and operator insertions.
• Figure 3: The plots of $\Delta S^{(2)}_A$ as functions of $t$ in the limit $\epsilon=0$. We chose $l=10$. The red and blue curve correspond to the operator ${\mathcal{O}}=\phi$ ($k=1$) in 6 and 4 dimension, respectively. The green graph describes the entropy for the operator $\mathcal{O}=:e^{i \alpha \phi}:+:e^{-i \alpha \phi}:$ in 2 dimension.
• Figure 4: This shows a schematic explanation for the time evolution of $\Delta S^{(n)}_A$ in terms of entangled pairs.
• Figure 5: The subsystem A is defined by the region $x_1>0, \tau=0$. Operators are inserted into the points $(r, \theta, x^i ),~ (r', \theta ', x^i )$.