Local quantum channels giving rise to quasi-local Gibbs states

TL;DR

This paper analyzes the steady states of local quantum channels, showing they are Gibbs states of quasi-local Hamiltonians with properties that depend on the interaction strength, and provides algorithms for expectation value computations.

Contribution

It establishes that steady states of local quantum channels are Gibbs states of quasi-local Hamiltonians with a structured expansion in interaction order.

Findings

Steady states are Gibbs states of quasi-local Hamiltonians.

Higher-order interaction terms decay exponentially under certain conditions.

Efficient classical algorithms exist for local observable expectation values.

Abstract

We study the steady-state properties of quantum channels with local Kraus operators. We consider a large family that consists of general ergodic 1-local (non-interacting) terms and general 2-local (interacting) terms. Physically, a repeated application of these channels can be seen as a simple model for the thermalization process of a many-body system. We study its steady state perturbatively, by interpolating between the 1-local and 2-local channels with a perturbation parameter $\epsilon$. We prove that under very general conditions, these states are Gibbs states of a quasi-local Hamiltonian. Expanding this Hamiltonian as a series in $\epsilon$, we show that the $k$'th order term corresponds to a $(k+1)$-local interaction term in the Hamiltonian, which follows the same interaction graph as the Kraus channel. We also prove a complementary result suggesting the existence of an interaction strength threshold, under which the total weight of the high-order terms in the Hamiltonian decays exponentially fast. For sufficiently small $\epsilon$, this implies both exponential decay of local correlation functions and a classical algorithm for computing expectation value of local observables in such steady states. Finally, we present numerical simulations of various channels that support our theoretical results.