Timelike Entanglement Entropy and TT¯ Deformation

In a previous work [Eur. Phys. J. C 80, 1117 (2020)] about the $T\overline{T}$ deformed ${\mathrm{CFT}}_{2}$, from the consistency requirement of the entanglement entropy, we found that, in addition to the usual spacelike entanglement entropy, a timelike entanglement entropy must be introduced and treated equally. Motivated by the recent explicit constructions of the timelike entanglement entropy and its bulk dual, we provide a comprehensive analysis of the timelike and spacelike entanglement entropies in the $T\overline{T}$ deformed finite size system and finite temperature system. The results confirm our prediction that in the finite size system only the timelike entanglement entropy receives a correction, while in the finite temperature system only the usual spacelike entanglement entropy gets a correction. These findings affirm the necessity of a complete measure including both spacelike and timelike entanglement entropies.