Area law for steady states of detailed-balance local Lindbladians

TL;DR

This paper demonstrates that steady states of certain local Lindbladians in quantum systems obey an area law for mutual information, enabling efficient tensor-network representations and analysis using Hamiltonian complexity tools.

Contribution

It establishes a mapping from Lindbladians satisfying detailed balance to local Hamiltonians, allowing the application of Hamiltonian complexity methods to analyze steady states.

Findings

Steady states obey an area law for mutual information.

Tensor-network representations of steady states can be efficiently constructed.

Mapping Lindbladians to Hamiltonians facilitates analysis of open quantum systems.

Abstract

We study steady-states of quantum Markovian processes whose evolution is described by local Lindbladians. We assume that the Lindbladian is gapped and satisfies quantum detailed balance with respect to a unique full-rank steady state $\sigma$. We show that under mild assumptions on the Lindbladian terms, which can be checked efficiently, the Lindbladian can be mapped to a local Hamiltonian on a doubled Hilbert space that has the same spectrum, and a ground state that is the vectorization of $\sigma^{1/2}$. Consequently, we can use Hamiltonian complexity tools to study the steady states of such open systems. In particular, we show an area-law in the mutual information for the steady state of such 1D systems, together with a tensor-network representation that can be found efficiently.