Limit theorems for empirical measures of interacting quantum systems in Wasserstein space

We prove fundamental properties of empirical measures induced by measurements performed on quantum $N$-body systems. More precisely, we consider measurements performed on the ground state of an interacting, trapped Bose gase in the Gross–Pitaevskii regime, known to exhibit Bose–Einstein condensation. For the corresponding empirical measure, we prove a weak law of large numbers with limit induced by the condensate wave function and characterize the fluctuations around through an appropriate central limit theorem.