TY - GEN
T1 - Quantum Wasserstein and Observability for Quantum Dynamics
AU - Golse, François
N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024/1/1
Y1 - 2024/1/1
N2 - 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 Rd×Rd with density operators on L2(Rd). 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].
AB - 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 Rd×Rd with density operators on L2(Rd). 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].
KW - Lions’ Hilbert Uniqueness Method (HUM)
KW - Observability
KW - Schrödinger equation
KW - Von Neumann equation
KW - Wasserstein distance
U2 - 10.1007/978-3-031-65195-3_6
DO - 10.1007/978-3-031-65195-3_6
M3 - Conference contribution
AN - SCOPUS:85206076882
SN - 9783031651946
T3 - Springer Proceedings in Mathematics and Statistics
SP - 129
EP - 148
BT - From Particle Systems to Partial Differential Equations - PSPDE X 2022
A2 - Carlen, Eric
A2 - Gonçalves, Patrícia
A2 - Soares, Ana Jacinta
PB - Springer
T2 - 10th International Conference on Particle Systems and Partial Differential Equations, PSPDE 2022
Y2 - 26 June 2022 through 30 June 2022
ER -