On holographic time-like entanglement entropy

In order to study the pseudo entropy of timelike subregions holographically, the previous smooth space-like extremal surface was recently generalized to mix space-like and time-like segments and the area becomes complex value. This paper finds that, if one tries to use such kind of piecewise smooth extremal surfaces to compute timelike entanglement entropy holographically, the complex area is not unique in general. We then generalize the original holographic proposal of spacelike entanglement entropy to pick up a unique area from all allowed "space-like+time-like" piecewise smooth extremal surfaces for a timelike subregion. We give some concrete examples to show the correctness of our proposal.