Embedding, simulation and consistency of $\mathcal{P}\mathcal{T}$ -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 $\mathcal{P}\mathcal{T}$-symmetry in $\mathcal{P}\mathcal{T}$-symmetric quantum theory. The embedding property was first studied by G"{u}nther and Samsonov to visualize the evolution of unbroken $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians on ${\mathbb{C}}^2$ by Hermitian Hamiltonians on ${\mathbb{C}}^4$. This paper investigates the properties of $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-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 $\cal PT$-symmetricHamiltonians\mathrm{.}Anobservationthattheunbroken$\cal PT$-symmetric quantum systems can be viewed as open systems in the conventional quantum mechanics accounts for the consistency of $\mathcal{P}\mathcal{T}$-symmetric quantum theory.