  In this article, we explore properties of pseudo entropy [1] in quantum field
theories and spin systems from several approaches. Pseudo entropy is a
generalization of entanglement entropy such that it depends on both an initial
and final state and has a clear gravity dual via the AdS/CFT. We numerically
analyze a class of free scalar field theories and the XY spin model. This
reveals the basic properties of pseudo entropy in many-body systems, namely,
the area law behavior, the saturation behavior, and the non-positivity of
difference between the pseudo entropy and averaged entanglement entropy in the
same quantum phase. In addition, our numerical analysis finds an example where
the strong subadditivity of pseudo entropy gets violated. Interestingly we find
that the non-positivity of the difference can be violated only if the initial
and final states belong to different quantum phases. We also present analytical
arguments which support these properties by both conformal field theoretic and
holographic calculations. When the initial and final states belong to different
topological phases, we expect a gapless mode localized along an interface,
which enhances the pseudo entropy leading to the violation of the
non-positivity of the difference. Moreover, we also compute the time evolution
of pseudo entropy after a global quench, were we observe that the imaginary
part of pseudo entropy shows interesting characteristc behavior.
