Characterization of generator sets yielding primitive SoS Lindbladians

Characterize the sets of Hermitian generators \(\{J_a\}\) beyond exactly solvable replacement and free-fermion models that satisfy the sufficient modular compatibility condition and thereby produce a primitive Lindbladian.

Background

The paper proves that the SoS superoperator constructed from modular annihilators is a primitive Lindbladian when the generators form an irreducible algebra and satisfy a specific modular compatibility condition. The condition has been demonstrated for the replacement Lindbladian and for the free-fermion family, but its applicability to broader interacting or otherwise non-exactly-solvable systems remains unresolved. A characterization of all generator families satisfying the condition would clarify when the modular-annihilator parent-Hamiltonian construction corresponds to a valid primitive dissipative dynamics.

References

Several directions remain open. First, our locality-improved Krylov bound relies on relatively coarse quasi-locality estimates and still grows polynomially with $\beta$. A more refined understanding of operator growth in the Krylov basis may lead to sharper bounds. Second, the sufficient condition in #1{thm:lindbladian} has so far been verified only for the replacement Lindbladian and the free-fermion family. Characterizing the generator sets ${J_a}$ that satisfy this condition beyond exactly solvable models is therefore an important direction for future work.

— Modular-Annihilator Parent Hamiltonians for Purified Gibbs States: Spectral Design and Controlled Approximation  (2608.30272 - Yi et al., 31 Aug 2026) in Section 5, “Conclusion and outlook”