  We show that a quantum state in a lattice spin (boson) system must be
long-range entangled if it has non-zero lattice momentum, i.e. if it is an
eigenstate of the translation symmetry with eigenvalue $e^{iP}\neq1$.
Equivalently, any state that can be connected with a non-zero momentum state
through a finite-depth local unitary transformation must also be long-range
entangled. The statement can also be generalized to fermion systems. Some
non-trivial consequences follow immediately from our theorem: (1) several
different types of Lieb-Schultz-Mattis-Oshikawa-Hastings (LSMOH) theorems,
including a previously unknown version involving only a discrete $\mathbb{Z}_n$
symmetry, can be derived in a simple manner from our result; (2) a gapped
topological order (in space dimension $d>1$) must weakly break translation
symmetry if one of its ground states on torus has nontrivial momentum - this
generalizes the familiar physics of Tao-Thouless; (3) our result provides
further evidence of the "smoothness" assumption widely used in the
classification of crystalline symmetry-protected topological (cSPT) phases.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics;High Energy Physics - Theory;Quantum Physics