Connecting global and local energy distributions in quantum spin models on a lattice

Local interactions in many-body quantum systems are generally non-commuting and consequently the Hamiltonian of a local region cannot be measured simultaneously with the global Hamiltonian. The connection between the probability distributions of measurement outcomes of the local and global Hamiltonians will depend on the angles between the diagonalizing bases of these two Hamiltonians. In this paper we characterize the relation between these two distributions. On one hand, we upperbound the probability of measuring an energy tau in a local region, if the global system is in a superposition of eigenstates with energies epsilon < tau. On the other hand, we bound the probability of measuring a global energy epsilon in a bipartite system that is in a tensor product of eigenstates of its two subsystems. Very roughly, we show that due to the local nature of the governing interactions, these distributions are identical to what one encounters in the commuting cases, up to exponentially small corrections. Finally, we use these bounds to study the spectrum of a locally truncated Hamiltonian, in which the energies of a contiguous region have been truncated above some threshold energy. We show that the lower part of the spectrum of this Hamiltonian is exponentially close to that of the original Hamiltonian. A restricted version of this result in 1D was a central building block in a recent improvement of the 1D area-law.