A classification of phases of bosonic quantum lattice systems in one dimension.

We study the entanglement properties of quantum phases of bosonic 1d lattice systems in infinite volume. We show that a ground state of any gapped local Hamiltonian is Short-Range Entangled: it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. We characterize Short-Range Entangled states in terms of decay properties of their Schmidt coefficients. If a Short-Range Entangled state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries. We show that in the case of a finite unitary symmetry G the only obstruction for the existence of a symmetry-preserving disentangler is an index valued in degree-2 cohomology of G. We show that two Short-Range Entangled states are in the same phase if and only if their indices coincide.