  Hamiltonian Learning (HL) is essential for validating quantum systems in
quantum computing. Not all Hamiltonians can be uniquely recovered from a steady
state. HL success depends on the Hamiltonian model and steady state. Here, we
analyze HL for a specific type of steady state composed of eigenstates with
degenerate mixing weight, making these Hamiltonian's eigenstates
indistinguishable. To overcome this challenge, we utilize the orthogonality
relationship between the eigenstate space and its complement space,
constructing the orthogonal space equation. By counting the number of linearly
independent equations derived from a steady state, we determine the
recoverability of a generic local Hamiltonian. Our scheme is applicable for
generic local Hamiltonians under various steady state, therefore offering a way
of measuring the degree to which a steady state characterizes a Hamiltonian.
