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We study the null space degeneracy of open quantum systems with multiplenon-Abelian, strong symmetries. By decomposing the Hilbert space representationof these symmetries into an irreducible representation involving the direct sumof multiple, commuting, invariant subspaces we derive a tight lower bound forthe stationary state degeneracy. We apply these results within the context ofopen quantum many-body systems, presenting three illustrative examples: afully-connected quantum network, the XXX Heisenberg model and the Hubbardmodel. We find that the derived bound, which scales at least cubically in thesystem size the $SU(2)$ symmetric cases, is often saturated. Moreover, our workprovides a theory for the systematic block-decomposition of a Liouvillian withnon-Abelian symmetries, reducing the computational difficulty involved indiagonalising these objects and exposing a natural, physical structure to thesteady states - which we observe in our examples.

Stationary State Degeneracy of Open Quantum Systems with Non-Abelian Symmetries