  In closed systems, the celebrated Lieb-Schultz-Mattis (LSM) theorem states
that a one-dimensional locally interacting half-integer spin chain with
translation and spin rotation symmetry cannot have a non-degenerate gapped
ground state. However, the applicability of this theorem is diminished when the
system interacts with a bath and loses its energy conservation. In this letter,
we propose that the LSM theorem can be revived in the entanglement Hamiltonian
when the coupling to bath renders the system short-range correlated.
Specifically, we argue that the entanglement spectrum cannot have a
non-degenerate minimum, isolated by a gap from other states. We further support
the results with numerical examples where a spin-$1/2$ system is coupled to
another spin-$3/2$ chain serving as the bath. Compared with the original LSM
theorem which primarily addresses UV--IR correspondence, our findings unveil
that the UV data and topological constraints also have a pivotal role in
shaping the entanglement in open quantum many-body systems.
