A Posteriori Certification Framework for Generalized Quantum Arimoto-Blahut Algorithms

TL;DR

This work introduces an a posteriori certification framework for generalized quantum Arimoto–Blahut (QAB) algorithms, addressing the difficulty of verifying global convergence with traditional a priori conditions. It proves a generalized global convergence theorem for convex objectives under a numerically verifiable inequality along iterates, enabling certification of global optimality and explicit suboptimality bounds from the computed trajectory. The framework is applied to compute the quantum relative entropy of channels, reformulating the problem as a convex minimization over input states and deriving efficient fixed-point updates that avoid large SDP formulations. Numerical experiments on qubit channels demonstrate rapid convergence, improved scalability, and reliable certification, highlighting practical advantages over SDP-based approaches in both memory and iteration requirements.

Abstract

The generalized quantum Arimoto--Blahut (QAB) algorithm is a powerful derivative-free iterative method in quantum information theory. A key obstacle to its broader use is that existing convergence guarantees typically rely on analytical conditions that are either overly restrictive or difficult to verify for concrete problems. We address this issue by introducing an a posteriori certification viewpoint: instead of requiring fully a priori verifiable assumptions, we provide convergence and error guarantees that can be validated directly from the iterates produced by the algorithm. Specifically, we prove a generalized global convergence theorem showing that, under convexity and a substantially weaker numerically verifiable condition, the QAB iteration converges to the global minimizer. This theorem yields a practical certification procedure: by checking explicit inequalities along the computed trajectory, one can certify global optimality and bound the suboptimality of the obtained value. As an application, we develop a certified iterative scheme for computing the quantum relative entropy of channels, a fundamental measure of distinguishability in quantum dynamics. This quantity is notoriously challenging to evaluate numerically: gradient-based methods are impeded by the complexity of matrix functions such as square roots and logarithms, while recent semidefinite programming approaches can become computationally and memory intensive at high precision. Our method avoids these bottlenecks by combining the QAB iteration with a posteriori certification, yielding an efficient and scalable algorithm. Numerical experiments demonstrate rapid convergence and improved scalability and adaptivity compared with SDP-based approaches.

Figures (6)

• Figure 1: Relative entropy of channels $D({\cal D}_{deph}\|{\cal D}_p)$ as a function of the depolarizing parameter $p$. The vertical axis shows the value of the $D({\cal D}_{deph}\|{\cal D}_p)$. The horizontal axis shows the value of $p$. The dephasing parameter is fixed as 0.4. The blue line represents the numerical results of our method and the red dot line represents the results shown in wilde2025sdp
• Figure 2: The maximum value and minimum value of ratio $D_{\Omega}(\rho^{(100)}\|\sigma)/D(\rho^{(100)}\|\sigma)$ for the convergent points in Fig. \ref{['fig:exp_4']}.$\rho^{(100)}$ denotes the convergent points we get in Fig. \ref{['fig:exp_4']}. Blue points denotes the maximum value of $D_{\Omega}(\rho^{(100)}\|\sigma)/D(\rho^{(100)}\|\sigma)$ among 10000 random states $\sigma$. Red points denotes the minimum value of $D_{\Omega}(\rho^{(100)}\|\sigma)/D(\rho^{(100)}\|\sigma)$ among 10000 random states $\sigma$
• Figure 3: The maximum value and minimum value of ratio $D_{\Omega}(\rho^{(100)}\|\rho^{(j)})/D(\rho^{(100)}\|\rho^{(j)})$ for the convergent points in Fig. \ref{['fig:exp_4']}.$\rho^{(100)}$ denotes the convergent points we get in Fig. \ref{['fig:exp_4']}. Blue points denotes the maximum value of $D_{\Omega}(\rho^{(100)}\|\rho^{(j)})/D(\rho^{(100)}\|\rho^{(j)})$ for $j=1, \ldots, 99$. Red points denotes the minimum value of $D_{\Omega}(\rho^{(100)}\|\rho^{(j)})/D(\rho^{(100)}\|\rho^{(j)})$ for $j=1, \ldots, 99$.
• Figure 4: The maximum value of ratio $D_{\Omega}(\rho^{(j+1)}\|\rho^{(j)})/D(\rho^{(j+1)}\|\rho^{(j)})$ across sequential state pairs ($j=1,..., 99$) for varying parameter $p$.
• Figure 5: The plot of the gap value between exact value and our numerical results considered in Fig. \ref{['fig:exp_4']}. The vertical axis shows the gap value. The horizontal axis shows the value of $p$.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Lemma 3
• proof