Wave Packets and the Quadratic Monge-Kantorovich Distance in Quantum Mechanics

Abstract

In this paper, we extend the upper and lower bounds for the "pseudo-distance" on quantum densities analogous to the quadratic Monge-Kantorovich(-Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165-205] to positive quantizations defined in terms of the family of phase space translates of a density operator, not necessarily of rank one as in the case of the Töplitz quantization. As a corollary, we prove that the uniform (for vanishing h) convergence rate for the mean-field limit of the N-particle Heisenberg equation holds for a much wider class of initial data than in [F. Golse, C. Mouhot, T. Paul, loc. cit.]. We also discuss the relevance of the pseudo-distance compared to the Schatten norms for the purpose of metrizing the set of quantum density operators in the semiclassical regime.

23 pages, no figure

Cited by in corpus (23)

• The Quantum Wasserstein Distance of Order 1
• Learning quantum data with the quantum Earth Mover's distance
• Quantum optimal transport with quantum channels
• Quantum Monge-Kantorovich problem and transport distance between density matrices
• Empirical Measures and Quantum Mechanics: Application to the Mean-Field Limit
• On the Convergence of Time Splitting Methods for Quantum Dynamics in the Semiclassical Regime
• Propagation of Moments and Semiclassical Limit from Hartree to Vlasov Equation
• Quantum Optimal Transport
• Monotonicity of the quantum 2-Wasserstein distance
• Optimal quantum channels
• Improving the speed of variational quantum algorithms for quantum error correction
• Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data
• Quantum Wasserstein distance based on an optimization over separable states
• Quantum Wasserstein isometries on the qubit state space
• The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
• On the metric property of quantum Wasserstein divergences
• Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance
• Quantum Wasserstein distance of order 1 between channels
• Classical shadows meet quantum optimal mass transport
• Mean-Field and Classical Limit for the N-Body Quantum Dynamics with Coulomb Interaction
• Isometries of the qubit state space with respect to quantum Wasserstein distances
• Quantum optimal transport is cheaper
• Time Dependent Quantum Perturbations Uniform in the Semiclassical Regime