Semidefinite programming for understanding limitations of Lindblad equations

We use semidefinite programming (SDP) to establish rigorous no-go theorems on the fundamental limitations of Lindblad equations in correctly modeling non-equilibrium steady states[cite: 2]. We tested the feasibility of constructing QMEs that satisfy physical consistency requirements and accurate steady-state properties, establishing rigorous no-go results for XXZ and XX spin chains[cite: 4]. This demonstrates that for most non-equilibrium and strongly coupled regimes, it is fundamentally impossible for a Markovian QME to satisfy local conservation laws while correctly reproducing the leading-order populations and coherences of the non-equilibrium steady state (NESS)[cite: 4].