Quantum Wasserstein and Observability for Quantum Dynamics

In recent years, there have been several extensions of various tools and methods of optimal transport to the quantum setting. In particular, a pseudometric analogous to the Wasserstein distance of exponent 2 has been defined in [F. Golse, T. Paul, Arch. Rational Mech. Anal. 223 (2017) 57–94] for the purpose of comparing probability densities defined on \(\textbf{R}^d\times \textbf{R}^d\) with density operators on \(L^2(\textbf{R}^d)\). This pseudometric is particularly convenient if one seeks a quantitative error estimate for the classical limit of quantum dynamics. In this talk, we explain how to use this tool in order to study the observability problem for the Schrödinger or the von Neumann equations. Our analysis of this problem uses the quantum analogue of the Wasserstein distance, together with a geometric condition on the classical trajectories corresponding to the quantum dynamics under a condition analogous to the Bardos-Lebeau-Rauch geometric condition for the exact controllability of the wave equation [C. Bardos, G. Lebeau, J. Rauch, SIAM J. Control Opti. 30 (1992) 1024–1065]. The material presented in this paper is a review of a series of joint works with T. Paul, especially [Math. Models Methods Appl. Sciences, 32 (2022) 941–963].