Determining non-Hermitian parent Hamiltonian from a single eigenstate

TL;DR

This paper introduces a method to determine non-Hermitian Hamiltonians from a single eigenstate using quantum covariance matrices, facilitating experimental Hamiltonian learning despite measurement errors.

Contribution

It demonstrates that a non-Hermitian Hamiltonian can be uniquely identified from a single eigenstate, simplifying the process for experimental systems.

Findings

Hamiltonian can be reconstructed from a single eigenstate.

The method is stable against measurement errors.

Applicable to experimental quantum systems.

Abstract

A quantum state for being an eigenstate of some local Hamiltonian should be constraint 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 favours 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.