  We systematically study the parity- and time-reversal (PT) symmetric
non-Hermitian version of a quantum network proposed in the paper of Christandl
et al. [Phys. Rev. Lett. 92, 187902 (2004)]. The nature of this model shows
that it is a paradigm to demonstrate the complex relationship between the
pseudo-Hermitian Hamiltonian and its Hermitian counterpart as well as a
candidate in the experimental realization to simulate PT-symmetry breaking. We
also show that this model allows a conditional perfect state transfer within
the unbroken PT-symmetry region but not an arbitrary one. This is due to the
fact that the evolution operator at a certain period is equivalent to the PT
operator for the real-valued wave function in the elaborate PT-symmetric
Hilbert space.
