Calculus of Variations and Geometric Measure Theory

D. Feliciangeli - A. Gerolin - L. Portinale

A Non-Commutative Entropic Optimal Transport Approach to Quantum Composite Systems at Positive Temperature

created by portinale on 22 Jun 2021
modified on 11 May 2023


Published Paper

Inserted: 22 jun 2021
Last Updated: 11 may 2023

Journal: Journal of Functional Analysis
Year: 2023

ArXiv: 2106.11217 PDF


This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on careful a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.