Quantum Thermodynamics and Semidefinite Programming: Regularization and Algorithms

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 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.