Description
Non-Hermitian Hamiltonians have biorthogonal quantum properties, defined through left and right eigenstates, that are not directly accessible in standard quantum measurements. Naimark dilation offers a route around this obstacle. It embeds a pseudo-Hermitian system into a larger Hermitian system with an ancilla, while preserving the energy spectrum. We first realize this protocol on quantum hardware for the zero-dimensional Yang-Lee model, and reproduce its biorthogonal expectation values. We then ask what the dilation implies for many-body states, and turn to the non-Hermitian Su-Schrieffer-Heeger model. The dilation requires a pseudo-Hermiticity operator, which is not unique. This freedom is immaterial in zero dimensions. On the lattice, however, we find that it controls both the locality of the enlarged Hamiltonian and the entanglement structure of its ground state. In particular, a similarity transformation also plays an essential role in constructing an enlarged Hamiltonian whose entanglement entropy follows the expected Cardy-Calabrese scaling. These results identify the pseudo-Hermiticity operator as a central ingredient in relating non-Hermitian systems to Hermitian many-body states. We also discuss the implications for connecting non-unitary and unitary conformal field theories at the level of quantum states.