Local quantum channels giving rise to quasi-local Gibbs states
Itai Arad, Raz Firanko, Omer Gurevich
TL;DR
This work analyzes open quantum systems driven by locally structured stochastic maps, showing that repeated application yields steady states that are Gibbs states of quasi-local Hamiltonians. By introducing a perturbative expansion in a locality parameter $\epsilon$ and leveraging a multivariate edge-parameter framework, the authors prove that the Gibbs Hamiltonian $H_G(\epsilon)=-\log\rho_\infty(\epsilon)$ decomposes as $H_G(\epsilon)=\sum_{k\ge0} \epsilon^k H_k$ with each $H_k$ being geometrically $(k{+}1)$-local; under plausible growth bounds, $H_G(\epsilon)$ is quasi-local for small $\epsilon$. They further derive a Heisenberg-picture perturbation expansion for local observables $A$, $\langle A\rangle=\sum_k \epsilon^k \mathrm{Tr}(\rho_0 A_k)$, with $A_k$ supported in the ball of radius $k$ and ${\|A_k\|}_\infty\le e^{6(k+\ell)}{\|A\|}_\infty$, ensuring a radius of convergence $\epsilon_0=1/e^6$ and enabling a quasi-polynomial classical algorithm to approximate local expectations in $D$-dimensional lattices. The authors prove exponential decay of correlations for small $\epsilon$ and provide numerical evidence across several channel models that the steady-state Gibbs Hamiltonian is indeed quasi-local. These results illuminate a route to Gibbs-state preparation via local quantum maps and suggest a complexity transition as $\epsilon$ approaches unity.
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 $ε$. We prove that under very general conditions, these states are Gibbs states of a quasi-local Hamiltonian. Expanding this Hamiltonian as a series in $ε$, 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 $ε$, 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.