Simulating broken $cal PT$-symmetric Hamiltonian systems by weak measurement

By embedding a $cal PT$-symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between $cal{PT}$-symmetric Hamiltonians and weak measurement theory. We show that the amplification effect in weak measurement on a conventional quantum system can be used to effectively simulate a local broken $cal PT$-symmetric Hamiltonian system, with the pre-selected state in the $cal PT$-symmetric Hamiltonian system and its post-selected state resident in the dilated Hamiltonian system.