We systematically study the parity- and time-reversal ($\mathcal{PT}$) 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 $\mathcal{PT}$-symmetry breaking. We also show that this model allows a conditional perfect state transfer within the unbroken $\mathcal{PT}$-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 $\mathcal{PT}$ operator for the real-valued wave function in the elaborate $\mathcal{PT}$-symmetric Hilbert space.

• Received 1 June 2011

DOI:https://doi.org/10.1103/PhysRevA.85.012106