🤖 AI Summary
This work addresses the lack of efficient numerical methods for computing the quantum channel relative entropy—a fundamental metric in quantum channel discrimination and resource theories. Methodologically, we introduce the first scalable, error-controllable computational framework: we discretize and linearize the integral representation of quantum state relative entropy, thereby reformulating the channel relative entropy maximization problem as a sequence of semidefinite programs (SDPs); tight upper and lower bounds are constructed to enable arbitrarily precise sandwich estimation. Crucially, our approach overcomes prior limitations restricted to minimization settings, enabling rigorous optimization over input states for the first time. Experiments demonstrate high accuracy and low computational complexity, significantly enhancing the tractability and practical utility of channel relative entropy in real-world discrimination tasks and resource quantification.
📝 Abstract
This paper introduces a method for calculating the quantum relative entropy of channels, an essential quantity in quantum channel discrimination and resource theories of quantum channels. By building on recent developments in the optimization of relative entropy for quantum states [Kossmann and Schwonnek, arXiv:2404.17016], we introduce a discretized linearization of the integral representation for the relative entropy for states, enabling us to handle maximization tasks of the relative entropy of a channel over input states. Our approach here extends previous work on minimizing relative entropy to the more complicated domain of maximization. It also provides efficiently computable upper and lower bounds that sandwich the true value with any desired precision, leading to a practical method for computing the relative entropy of channels.