In this 48-minute lecture, Francois Golse from École Polytechnique explores the observability problem in quantum mechanics through the lens of Wasserstein distance. Recorded at IPAM's Dynamics of Density Operators Workshop on April 30, 2025, discover an innovative approach to the observability problem for the Schrödinger equation and quantum Liouville (von Neumann) equation based on propagation estimates for a quantum analogue of the Wasserstein distance of order two. The presentation covers collaborative work with Thierry Paul, offering valuable insights for researchers and students interested in quantum mechanics, mathematical physics, and differential equations.