Circulant quantum channels and its applications
Bing Xie, Lin Zhang
TL;DR
This work introduces circulant quantum channels as a structured subclass of mixed-permutation channels and proves that their outputs are exactly circulant matrices, enabling a unified approach to Bargmann invariants and coherence estimation. It characterizes the channel's fixed-point space, spectrum, and unitality, and shows the uniform-weight case is entanglement-breaking via a separable Choi operator, with a complete EB criterion across the weight family. The authors demonstrate tighter lower bounds for $\ell_p$-norm coherence ($p=1,2$) compared to broader channels and provide a Bargmann invariant reformulation that equates $n$-th order invariants with circulant Gram matrices. They extend the framework to bipartite systems, showing that local circulant channels can completely destroy entanglement in small-dimension cases, while nonuniform weights can preserve entanglement. Overall, the Circulant quantum channel offers a concise, symmetry-driven toolkit for quantum coherence, Bargmann invariants, and entanglement analysis with potential for broader circulant-channel families.
Abstract
This note introduces a family of circulant quantum channels -- a subclass of the mixed-permutation channels -- and investigates its key structural and operational properties. We show that the image of the circulant quantum channel is precisely the set of circulant matrices. This characterization facilitates the analysis of arbitrary $n$-th order Bargmann invariants. Furthermore, we prove that the channel is entanglement-breaking, implying a substantially reduced resource cost for erasing quantum correlations compared to a general mixed-permutation channel. Applications of this channel are also discussed, including the derivation of tighter lower bounds for $\ell_p$-norm coherence and a characterization of its action in bipartite systems.