CP-preserving channels

Abstract:Completely positive (CP) matrices are ubiquitous in modern science and technology with applications in optimization, graph theory, and quantum entanglement. Recently, Johnston \emph{et al.} [Linear Algebra and its Applications, 2022] have cast CP matrices into the framework of quantum resource theories, where CP states serve as free states and CP-preserving channels act as free operations. This work addresses several questions raised in their work. Specifically, we provide the necessary and sufficient conditions of CP-preserving channels in small dimensions, which are necessary in higher dimensions, and discuss the resource quantification via the trace distance of non-negativity. By constructing an explicit counterexample, we demonstrate that the trace-distance measure of non-negativity violates strong monotonicity. We also provide an alternative proof that every CPDNN channel $\Phi:\MM_n\to \MM_2$ is CPCP. Additionally, we show that any unital CPDNN map $\Phi:\MM_2\to \MM_n$ is also CPCP.

13 pages (single column); comments/suggestions welcome!

arXiv:2607.24682 [quant-ph]

(or arXiv:2607.24682v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.24682 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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