An open quantum system — one coupled to an environment — evolves under the Lindblad master equation. The generator of this evolution, the Liouvillian, is a non-Hermitian superoperator whose eigenvalues are complex: real parts give decay rates, imaginary parts give oscillation frequencies. The topology of the Liouvillian spectrum — specifically, whether its eigenvalues wind nontrivially around the origin in the complex plane — determines non-equilibrium phenomena like the non-Hermitian skin effect, where eigenstates pile up at boundaries.
The Hamiltonian of the system, meanwhile, has its own topology: band invariants, winding numbers, Chern numbers. In a closed system, these classify gapped phases. The question for open systems is whether the Hamiltonian topology survives the dissipation, or whether the environment rewrites the topological classification entirely.
Long, Yang, Mu, and Li (arXiv 2602.22323, February 2026) show that when the Hamiltonian and the quantum jump operators share the same chiral symmetry, the Hamiltonian band topology directly controls the Liouvillian spectral winding. The topology isn't merely preserved — it's inherited. The dissipative system doesn't develop its own independent topological structure; it takes its topological character from the coherent Hamiltonian.
The mechanism is exact. For one-dimensional dissipative lattice models, the Liouvillian spectrum is solved analytically. The spectral winding number of the Liouvillian equals a function of the Hamiltonian band topology. Changing the Hamiltonian's topological phase changes the Liouvillian's winding, and vice versa: if the Hamiltonian is topologically trivial, the Liouvillian is too, regardless of how strong the dissipation is.
Lattice parity adds another layer. Even and odd lattice sizes produce different bulk-boundary correspondences in the steady state — the non-Hermitian skin effect either appears or doesn't depending on whether the lattice has an even or odd number of sites. The parity of the geometry interacts with the inherited topology to determine the physical behavior.
The result provides a control mechanism. Rather than treating dissipation as a destroyer of topological order, the symmetry protection allows the Hamiltonian to function as a dial: tune the band topology, and the dissipative dynamics follows. The environment doesn't erase the topology. The symmetry ensures it can't.