Certifying steady-state properties of open quantum systems

TL;DR

This paper introduces a scalable semi-definite programming approach to certify bounds on steady-state properties of open quantum systems, providing guaranteed estimates where previous methods only offered approximate results.

Contribution

The work presents the first general numerical tool for certifying steady-state observables in open quantum systems, applicable to complex many-body models.

Findings

Efficiently computes certified bounds for large many-body systems.

Outperforms tensor-network methods by providing guarantees.

Applicable to both equilibrium and nonequilibrium steady-states.

Abstract

Estimating the steady-state properties of open many-body quantum systems is a fundamental challenge in quantum science and technologies. In this work, we present a scalable approach based on semi-definite programming to derive certified bounds on the expectation value of an arbitrary observable in the steady state of Lindbladian dynamics. We illustrate our method on a series of many-body systems, including paradigmatic spin-1/2 chains and two-dimensional ladders, considering both equilibrium and nonequilibrium steady-states. We benchmark our method with state-of-the-art tensor-network approaches that, unlike our method, are only able to provide estimates, with no guarantee, on steady-state quantities. For the tested models, only modest computational effort is needed to obtain certified non-trivial bounds for system sizes intractable by exact methods. Our method introduces the first general numerical tool for bounding steady-state properties of open quantum dynamics, opening a new avenue in the understanding of stable configurations in many-body systems.