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

TL;DR

This paper introduces a novel non-commutative entropy regularized optimal transport framework for quantum systems at positive temperature, enabling new computational methods and convergence guarantees in quantum chemistry.

Contribution

It develops a non-commutative optimal transport approach for quantum systems, including a Sinkhorn algorithm with proven convergence and robustness, extending to fermionic and bosonic symmetries.

Findings

Established a non-commutative Sinkhorn algorithm with convergence guarantees

Connected quantum many-body problems with optimal transport theory

Extended methods to systems with symmetry constraints

Abstract

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.