  In conventional quantum mechanics, quantum no-deleting and no-cloning
theorems indicate that two different and nonorthogonal states cannot be
perfectly and deterministically deleted and cloned, respectively. Here, we
investigate the quantum deleting and cloning in a pseudo-unitary system. We
first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a
two-qubit system. By using the pseudo-unitary operators generated from this
pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a
class of two different and nonorthogonal states, and it can be generalized to
arbitrary two different and nonorthogonal pure qubit states. Furthermore, state
discrimination, which is strongly related to quantum no-cloning theorem, is
also discussed. Last but not least, we simulate the pseudo-unitary operators in
conventional quantum mechanics with post-selection, and obtain the success
probability of simulations. Pseudo-unitary operators are implemented with a
limited efficiency due to the post-selections. Thus, the success probabilities
of deleting and cloning in the simulation by conventional quantum mechanics are
less than unity, which maintain the quantum no-deleting and no-cloning
theorems.
