  We introduce a quantity, called pseudo entropy, as a generalization of
entanglement entropy via post-selection. In the AdS/CFT correspondence, this
quantity is dual to areas of minimal area surfaces in time-dependent Euclidean
spaces which are asymptotically AdS. We study its basic properties and
classifications in qubit systems. In specific examples, we provide a quantum
information theoretic meaning of this new quantity as an averaged number of
Bell pairs when the post-selection is performed. We also present properties of
the pseudo entropy for random states. We then calculate the pseudo entropy in
the presence of local operator excitations for both the two dimensional free
massless scalar CFT and two dimensional holographic CFTs. We find a general
property in CFTs that the pseudo entropy is highly reduced when the local
operators get closer to the boundary of the subsystem. We also compute the
holographic pseudo entropy for a Janus solution, dual to an exactly marginal
perturbation of a two dimensional CFT and find its agreement with a
perturbative calculation in the dual CFT. We show the linearity property holds
for holographic states, where the holographic pseudo entropy coincides with a
weak value of the area operator. Finally, we propose a mixed state
generalization of pseudo entropy and give its gravity dual.
