A NonCommutative Entropic Optimal Transport Approach to Quantum Composite Systems at Positive Temperature
Abstract
This paper establishes new connections between manybody quantum systems, Onebody Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the groundstate energy of a finite dimensional composite quantum system at positive temperature as a noncommutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dualprimal solutions in such noncommutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the manyelectron ground state energy can be computed via a noncommutative 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.11217
 Bibcode:
 2021arXiv210611217F
 Keywords:

 Mathematical Physics;
 Mathematics  Optimization and Control