Explore a lecture on density matrix embedding theory (DMET) and its application to quantum systems. Delve into the challenges of achieving exact matching in DMET calculations and the proposed solution using an augmented Lagrangian method. Learn about the alm-DMET approach, which relaxes the non-interacting pure-state v-representability assumption and allows for pure states with arbitrary occupation profiles. Discover how this method improves numerical accuracy compared to conventional self-consistent DMET techniques and its potential impact on treating strong correlation effects in large quantum systems.