  The Lieb-Schultz-Mattis (LSM) theorem provides a general constraint on
quantum many-body systems and plays a significant role in the Haldane gap
phenomena and topological phases of matter. Here, we extend the LSM theorem to
open quantum systems and establish a general theorem that restricts the steady
state and spectral gap of Liouvillians based solely on symmetry. Specifically,
we demonstrate that the unique gapped steady state is prohibited when
translation invariance and U (1) symmetry are simultaneously present for
noninteger filling numbers. As an illustrative example, we find that no
dissipative gap is open in the spin-1/2 dissipative Heisenberg model while a
dissipative gap can be open in the spin-1 counterpart -- an analog of the
Haldane gap phenomena in open quantum systems. Furthermore, we show that the
LSM constraint manifests itself in a quantum anomaly of the dissipative form
factor of Liouvillians. We also find the LSM constraints due to symmetry
intrinsic to open quantum systems, such as Kubo-Martin-Schwinger symmetry. Our
work leads to a unified understanding of phases and phenomena in open quantum
systems.
