Universal relation between thermal entropy and entanglement entropy in CFT

Inspired by the holographic computation of large interval entanglement entropy of two dimensional conformal field theory at high temperature, it was proposed that the thermal entropy is related to the entanglement entropy as $S_{th}={\displaystyle \underset{l\to 0}{\lim }(S_{EE}(R-l)-S_{EE}(l))}$. In this letter, we prove this relation for 2D CFT with discrete spectrum in two different ways. Moreover we discuss this relation for a 2D noncompact free scalar, which is a gapless CFT with continuous spectrum. We show that it could be recovered, after appropriately regularizing the theory.