Determining the non-Hermitian parent Hamiltonian from a single eigenstate

A quantum state for being an eigenstate of some local Hamiltonian should be constrained by zero energy variance and consequently, the constraint is rather strong that a single eigenstate may uniquely determine the Hamiltonian. For non-Hermitian systems, it is natural to expect that determining the Hamiltonian requires a pair of both left and right eigenstates. Here, we observe that it can be sufficient to determine a non-Hermitian Hamiltonian from a single right or left eigenstate. Our approach is based on the quantum covariance matrix, where the solution of Hamiltonian corresponds to the complex null vector. Our scheme favors non-Hermitian Hamiltonian learning on experimental quantum systems, as only the right eigenstates there can be accessed. Furthermore, we use numerical simulations to examine the effects of measurement errors and show the stability of our scheme.