Quantum channel relative entropy either finite or never attained
Finite attainment and efficient computation of regularized quantum channel relative entropy
Information Theory
Summary
Quantum channel relative entropy measures how different two quantum communication methods are but can behave unpredictably over multiple uses. The authors found a clear split: this measure is either achieved right away with one use or can never be exactly reached with any finite number of uses. They also discovered a simple test that works for certain channels to determine which case applies. Additionally, they show how to efficiently approximate this measure with a known number of channel copies or computational steps. This sharpens understanding of when and how this quantum measure can be computed in practice.
What this means in practice
- •For quantum communication engineers: Determine if a quantum communication channel’s distinguishability measure can be computed exactly with limited resources or requires approximation strategies.
- •For quantum algorithm designers: Use efficient algorithms from the paper to approximate regularized quantum channel relative entropy within certified polynomial time for improved quantum protocol design.
Authors
Yiwei Xu, Kun Fang
Abstract
Quantum channel relative entropy can be nonadditive, making its regularization essential for asymptotic analysis. We study when the regularized value is attained at finite blocklength and how efficiently it can be approximated. For the first question, we establish a sharp dichotomy: the regularized value is either attained already at the single-copy level, implying additivity at every blocklength, or is never attained at any finite blocklength. Notably, a simple two-copy additivity test distinguishes between these alternatives for qubit channels and for channel pairs in arbitrary dimensions admitting an interior optimizer. We further show that both the interiority condition and finite nonattainment are generic, holding for almost every channel pair. For the second question, we establish explicit convergence rates for approximating the regularized channel relative entropy. For fixed channel dimensions, $\widetilde O(\varepsilon^{-3/2})$ channel copies suffice to approximate the regularized value to additive accuracy $\varepsilon$. Alternatively, a semidefinite hierarchy achieves the same accuracy at level $\widetilde O(\varepsilon^{-2})$, yielding a certified polynomial-time approximation. Together, these results reveal the intriguing structure and generic behavior of regularized channel relative entropy, sharpening our understanding of both its analytical and computational aspects.