  The pseudo entropy is a promising recent generalization of the entanglement
entropy to the situations in which both the initial and final state are
involved, with the density matrix promoted to the transition matrix. However,
in contrast to the non-Hermiticity of the generic transition matrix, the
holographic pseudo entropy formulated via the Euclidean AdS/CFT turns out to be
always real-valued, which potentially conceals the crucial natures of this
novel quantity. In this note, we make the first attempt to formulate a
real-time prescription for computations to incorporate naturally the pseudo
entropy, as a generally complex-valued entanglement measure, into the AdS/CFT
context. It is then conjectured that the holographic pseudo entropy is dual to
the extremal codimension-2 area surface in the generally time-dependent
Lorentzian asymptotically AdS spacetime, but may also receive imaginary
contribution from the regularized extrinsic curvature term of the area surface,
which is not included in the covariant holographic entanglement entropy. In
this real-time prescription, the holographic pseudo entropy can be considered
as a generalization of the covariant holographic entanglement entropy, as well.
