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.Comment: 23 pages, 2 figures. A new version with tigheter bounds and a re-written introductio

Local reversibility and entanglement structure of many-body ground states

The low-temperature physics of quantum many-body systems is largely governed by the structure of their ground states. Minimizing the energy of local interactions, ground states often reflect strong properties of locality such as the area law for entanglement entropy and the exponential decay of correlations between spatially separated observables. In this letter we present a novel characterization of locality in quantum states, which we call `local reversibility'. It characterizes the type of operations that are needed to reverse the action of a general disturbance on the state. We prove that unique ground states of gapped local Hamiltonian are locally reversible. This way, we identify new fundamental features of many-body ground states, which cannot be derived from the aforementioned properties. We use local reversibility to distinguish between states enjoying microscopic and macroscopic quantum phenomena. To demonstrate the potential of our approach, we prove specific properties of ground states, which are relevant both to critical and non-critical theories.Comment: 12 revtex pages, 2 pdf figs; minor changes, typos corrected. To be published in Quantum Science and Technolog