Simulating broken <i>PT</i>-symmetric Hamiltonian systems by weak measurement

By embedding a PT-symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between PT-symmetric quantum theory and weak measurement theory. We show that the weak measurement can give rise to the inner product structure of PT-symmetric systems, with the pre-selected state and its post-selected state resident in the dilated conventional system. Typically in quantum information theory, by projecting out the irrelevant degrees and projecting onto the subspace, even local broken PT-symmetric Hamiltonian systems can be effectively simulated by this weak measurement paradigm.