Quantum thermodynamics and semidefinite programming: regularization and algorithms

TL;DR

This paper develops a mathematical framework for quantum thermodynamics problems constrained by measurement outcomes, introduces regularization techniques, and explores computational algorithms with convergence analysis.

Contribution

It provides a general regularization framework for quantum thermodynamics variational problems and analyzes dual formulations and algorithm convergence.

Findings

Established existence and characterization of maximizers.

Analyzed zero-temperature limit behavior.

Addressed algorithm convergence in quantum state tomography.

Abstract

We investigate variational problems in quantum thermodynamics at positive temperature, in which admissible states are constrained by prescribed outcomes of a finite set of measurements. We solve a problem raised by the recent work [Liu, Minervini, Patel, Wilde; arXiv:2505.04514 - Section C] and develop a general mathematical setup which allows a broad class of possible regularizations. Employing methods inspired by non-commutative optimal transport, we analyze the dual formulation of the problem, study the existence and characterization of maximizers, and investigate the qualitative behavior of the model in the zero-temperature limit. In the second part, we tailor this framework to quantum state tomography and quantum optimal transport. Finally, we address computational aspects, with particular attention to the convergence of algorithms in selected cases.