CP-preserving channels

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.