Generic Local Quantum Channels Give Rise to Quasi Local Gibbs states

We consider a wide family of quantum channels that are described by local Kraus maps. These maps consist of 1-local (non-interacting) and 2-local (interacting) terms, which can be easily implemented on a quantum computer. Physically, a repeated application of these channels can be seen as a simple model for the thermalization process of a many-body system, and it is therefore interesting to understand the steady states of such systems. By taking the overall strength of the 2-local terms as a pertubation parameter, we develop a complete perturbation theory for such steady states. We prove that that under very general conditions, these states are Gibbs states (a thermal equilibrium state of a system) of a ‘quasi-local Hamiltonian’, in which the k'th order term in the perturbation corresponds to (k+1)-local term in the Hamiltonian. We also prove a complementary result suggesting that for sufficiently small interaction strength, the total weight of the non-local terms in the Hamiltonian decays exponentially fast. This result also implies an efficient classical algorithm for computing the expectation value of local observables in such steady states. Finally, we also present numerical simulations of various channels that support our theoretical claims.