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

Generally, the local interactions in a many-body quantum spin system on a lattice do not commute with each other. Consequently, the Hamiltonian of a local region will generally not commute with that of the entire system, and so the two cannot be measured simultaneously. 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 case, up to some 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 $\tau$. 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.