Quantum thermodynamics and semidefinite programming: regularization and algorithms

TL;DR

A general mathematical setup is developed which allows a broad class of possible regularizations in variational problems in quantum thermodynamics at positive temperature and investigates the qualitative behavior of the model in the zero-temperature limit.

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 \cite[Section C]{MarkWilde} 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.

Figures (7)

• Figure 1: Quantum Tomography problem instance QT1. Graph of the $L_2$-norm of the gradient of the dual functional ${\rm D}_\varepsilon$ in Eq. \ref{['intro:maindual']} for the $\rho_1$ state; see \ref{['eq:QT1_inst']} in section \ref{['sec:numerics']}. Panels (a)--(f) display the iteration trajectories for different values of the regularization parameter $\varepsilon$ for both von Neumann (orange) and quadratic regularization (blue).
• Figure 2: Quantum Tomography problem instance QT2. Graph of the $L_2$-norm of the gradient of the dual functional ${\rm D}_\varepsilon$ in Eq. \ref{['intro:maindual']} for the $\rho_2$ state; see \ref{['eq:QT2_inst']} in section \ref{['sec:numerics']}. Panels (a)--(f) display the iteration trajectories for different values of the regularization parameter $\varepsilon$ for both von Neumann (orange) and quadratic regularization (blue).
• Figure 3: Quantum Tomography problem instance QT3. Graph of the $L_2$-norm of the gradient of the dual functional ${\rm D}_\varepsilon$ in Eq. \ref{['intro:maindual']} for the $\rho_3$ state; see \ref{['eq:QT3_inst']} in section \ref{['sec:numerics']}. Panels (a)--(f) display the iteration trajectories for different values of the regularization parameter $\varepsilon$ for both von Neumann (orange) and quadratic regularization (blue).
• Figure 4: Quantum Optimal Transport problem instance QWD. Graph of the absolute difference between the dual functional at iteration $k$${\rm D}_\varepsilon(\bm \alpha_k)$ in Eq. \ref{['intro:maindual']} and true optimal value $d^*$. Panels (a)--(c) display the iteration trajectories for different values of the regularization parameter $\varepsilon \in \{10^{4}, 10^{3}, 10^{1}\}$ for both von Neumann (orange) and quadratic regularization (blue).
• Figure 5: Quantum Optimal Transport problem instance QWD. Graph of the absolute difference between the dual functional at iteration $k$${\rm D}_\varepsilon(\bm \alpha_k)$ in Eq. \ref{['intro:maindual']} and true optimal value $d^*$. Panels (a)--(c) display the iteration trajectories for different values of the regularization parameter $\varepsilon \in \{10^{-2}, 10^{-6}, 10^{-9}\}$ for both von Neumann (orange) and quadratic regularization (blue).

Theorems & Definitions (36)

• Definition 2.1: Primal problem
• Remark 2.2: Boundedness of $\mathop{\mathrm{\rm{Adm}(\mathbf{Q},\mathbf{q})}}\nolimits$
• Proposition 2.3: Non-emptiness of $\mathop{\mathrm{\underline{\rm{Adm}}(\rm{\mathbf{Q}}, \rm{\mathbf{q}})}}\nolimits$
• Definition 2.4: Dual problem
• Theorem 2.5: Duality and optimizers
• Remark 2.6: Duality
• Theorem 2.7: Convergence as $\varepsilon \to 0$ and duality at zero temperature
• Proposition 3.1: Legendre's transform for functional calculus
• proof
• Remark 3.2: Uniqueness