Simulating Broken $\mathcal{PT}$-Symmetric Hamiltonian Systems by Weak Measurement

By embedding a $\mathcal{PT}$-symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between $\mathcal{PT}$-symmetric quantum theory and weak measurement theory. We show that the weak measurement can give rise to the inner product structure of $\mathcal{PT}$-symmetric systems, with the preselected state and its postselected 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 $\mathcal{PT}$-symmetric Hamiltonian systems can be effectively simulated by this weak measurement paradigm.