Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems

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 $\sigma$ 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 $exp(-((e-f)^2\sigma)^{\frac{1}{D+1}}/n^{\frac{1}{D+1}}D\sigma)$. This bound holds whenever $|e-f|>2^{D}\sqrt{n\sigma}$. 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 $exp(-(e-f)^2/nm^2)$. This bound holds whenever $|e-f|>m\sqrt{n}$.