Understanding scaling laws of entropies and mutual information has benefited studying correlated states of matter, quantum field theory, and gravity. Measuring von Neumann entropy experimentally in quantum many-body systems is challenging. In my talk, I will present our measurements of von Neumann entropy of subsystems in an ultracold atom simulator of Klein-Gordon field theory, verifying a foundational property of equilibrium states of gapped quantum many-body systems - the area law of quantum mutual information (Tajik, M. et al. Nat. Phys. 2023). I will also discuss the effect of temperature and subsystem separation on mutual information. Finally, I will address the challenges of measuring entanglement in many-body systems.