Abstract:

We consider the problem of determining the energy distribution of quantum states that satisfy exponential decay of correlation and product states, with respect to a quantum local Hamiltonian on a spin lattice. For a quantum state on a D-dimensional lattice that has correlation length σ and has average energy e with respect to a given local Hamiltonian (with n local terms, each of which has norm at most 1), we show that the overlap of this state with eigenspace of energy f is at most . This bound holds whenever . Thus, on a one-dimensional lattice, the tail of the energy distribution decays exponentially with the energy. For product states, we improve above result to obtain a Gaussian decay in energy, even for quantum spin systems without an underlying lattice structure. Given a product state on a collection of spins which has average energy e with respect to a local Hamiltonian (with n local terms and each local term overlapping with at most m other local terms), we show that the overlap of this state with eigenspace of energy f is at most . This bound holds whenever .