Conserved Quantities from Entanglement Hamiltonian

We show that the subregion entanglement Hamiltonians of excited eigenstates of a quantum many-body system are approximately linear combinations of subregionally (quasi)local approximate conserved quantities. By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigen-operators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is cut off around the order of subregion volume. In the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues decay roughly in power law if the system is integrable, with the exponent depending on if the eigenstates are extended (many-body localized). For fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, the eigen-operators of which correspond to the identity and the subregion Hamiltonian.