Generality of the implication from irreducibility to trivial decoherence-free algebra

Establish whether irreducibility of an arbitrary quantum Markov semigroup implies that its decoherence-free subalgebra is trivial, namely equal to the scalar algebra C1, beyond the classes for which this implication is already known.

Background

For Gaussian quantum Markov semigroups, the paper proves that irreducibility implies triviality of the decoherence-free subalgebra. It notes that the same implication is known for uniformly continuous quantum Markov semigroups admitting a faithful normal invariant state.

The authors explicitly state that the implication is not known in full generality for quantum Markov semigroups. This is a broader unresolved question than the GQMS-specific result established in the paper.

References

Finally, we remark that our results show that irreducibility implies that the decoherence-free subalgebra is trivial (Corollary \ref{coro:irrimplnt}), which is a known fact for uniformly continuous semigroups admitting a faithful normal invariant state (see, for instance, Proposition 4.3 in ), but it is not known to hold in full generality.

— Irreducibility and regularisation properties of Gaussian quantum Markov semigroups  (2609.01547 - Fagnola et al., 1 Sep 2026) in Section 1, Introduction