  The notion of non-Hermitian PT symmetric quantum theory has recently been
generalized to the gauge/gravity duality. We study the evolution of such
non-Hermitian holographic field theories when the couplings are varied with
time with particular emphasis on the question non-unitary time vs. unitary time
evolution. We show that a non-unitary time evolution in the dual quantum theory
corresponds to a violation of the Null Energy Condition (NEC) in the bulk of
the asymptotically AdS spacetime. We find that upon varying the non-Hermitian
coupling the horizon of a bulk AdS black hole shrinks. On the other hand
varying the Hermitian coupling in the presence of a constant non-Hermitian
coupling still violates the NEC but results in a growing horizon. We also show
that by introducing a non-Hermitian gauge field the time evolution can be made
unitary, e.g. the NEC in the bulk is obeyed, and an exactly equivalent purely
Hermitian description can be given.
