Embedding, simulation and consistency of $\cal PT$ -symmetric quantum Theory

A physical requirement on the Hamiltonian operator in quantum mechanics is that it must generate real energy spectrum and unitary time evolution. While the Hamiltonians are Dirac Hermitian in conventional quantum mechanics, they observe $\cal PT$-symmetry in $\cal PT$-symmetric quantum theory. The embedding property was first studied by G\"{u}nther and Samsonov to visualize the evolution of unbroken $\cal PT$-symmetric Hamiltonians on $\mathbb C^2$ by Hermitian Hamiltonians on $\mathbb C^4$. This paper investigates the properties of $\cal PT$-symmetric quantum systems including the embedding property. We provide a full characterization of the embedding property in the general case and show that only unbroken $\cal PT$-symmetric quantum systems admit this property in a finite dimensional space. Furthermore, utilizing this property, we establish a physically realizable simulation process of the unbroken $\cal PT$-symmetric Hamiltonians. An observation that the unbroken $\cal PT$-symmetric quantum systems can be viewed as open systems in the conventional quantum mechanics accounts for the consistency of $\cal PT$-symmetric quantum theory.