  We systematically study a Kitaev chain with imbalanced pair creation and
annihilation, which is introduced by non-Hermitian pairing terms. Exact phase
diagram shows that the topological phase is still robust under the influence of
the conditional imbalance. The gapped phases are characterized by a topological
invariant, the extended Zak phase, which is defined by the biorthonormal inner
product. Such phases are destroyed at the points where the coalescence of
groundstates occur, associating with the time-reversal symmetry breaking. We
find that the Majorana edge modes also exist for the open chain within unbroken
time-reversal symmetric region, demonstrating the bulk-edge correspondence in
such a non-Hermitian system.
