Local Certification of Many-Body Steady States

We present a relaxation-based method to bound expectation values on the steady state of dissipative many-body quantum systems described by master equations of the Lindblad form. Instead of targeting to represent the entire state, we promote the reduced density matrices to our variables and enforce the constraints that are imposed on them by consistency with a global steady state. The resulting constraints have the form of a semidefinite program, which allows us to efficiently bound the values a given expectation value can take in the steady state. Our results show fast convergence of the bounds with the size of the reduced density matrices, giving very competitive predictions for the steady state of several one- and two-dimensional models for an arbitrary number of particles.

💡 Research Summary

The paper introduces a relaxation‑based framework for rigorously bounding expectation values of local observables in the steady states of open many‑body quantum systems governed by Lindblad master equations. Instead of attempting to reconstruct the full density matrix, which scales exponentially with system size, the authors promote reduced density matrices of k contiguous sites, ρ(k), to the optimization variables. These reduced states must satisfy three classes of constraints: (i) positivity and unit trace, (ii) translation‑invariance (for translationally invariant systems) expressed as equality of partial traces, and (iii) consistency with the global steady‑state condition L(ρ_s)=0. The latter is translated into linear constraints on ρ(k) by exploiting the locality of the Lindbladian: the adjoint Lindbladian L† acting on an operator supported on k‑2 sites yields an operator supported on at most k sites, so Tr