Anshu, Anurag2024-02-072016-08-08Anshu, Anurag. "Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems." New J. Phys. 18, no. 8 (2016): 083011. DOI: 10.1088/1367-2630/18/8/0830111367-2630https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37377732We 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 exp(−((e−f)2σ)1D+1/n1D+1Dσ). This bound holds whenever |e−f|>2Dnσ‾‾‾√. 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/nm2). This bound holds whenever |e−f|>mn‾√.en-USGeneral Physics and AstronomyJournal Article2024-02-0710.1088/1367-2630/18/8/083011