Conserved quantities from entanglement, and timelike entanglement entropy

In the first part, we show that the subregion entanglement Hamiltonians of excited eigenstates of a many-body system are linear combinations of subregionally (quasi)local approximate conserved quantities, which can be obtained by diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM). For free fermions, EHSM gives a finite number of conserved quantities. In the second part, we generalize the entanglement entropy to timelike subregions, and give the formula for free fermions. For 1D lattice free fermions, the zero-temperature entanglement entropy in a discrete set of times in a time-direction interval stabilizes to the Cardy formula of chiral fermions when the time separation is below a critical value t0, and t0 is given by 2pi divided by the fermion bandwidth. We conjecture the time t0 is an upper bound for consecutive local observations to retrieve information from a quantum state.