Fig. 2: Quasi-locality of the maps.

For a lattice Hamiltonian, our Lindbladian is a sum over quasi-local terms \({{\mathcal{L}}}^{a}\) surrounding each jump A^a with radius approximately β, with an exponentially decaying tail controlled by the Lieb–Robinson bounds. This locality remains to hold for the parent Hamiltonian \({\mathcal{H}}=\sum _{a}{{\mathcal{H}}}^{a}\) of the purified Gibbs state \(| \sqrt{{\rho }_{\beta }}\rangle \) defined on two copies of the system.