AI 中文总结
研究保CP信道相关问题，给出小维度下保CP信道充要条件，通过构造反例证明非负性迹距离度量违反强单调性，还证明了特定CPDNN信道和幺正CPDNN映射是CPCP。
AI 中文摘要
完全正定（CP）矩阵在现代科技中无处不在，应用于优化、图论和量子纠缠等领域。最近，约翰斯顿等人[《线性代数及其应用》，2022年]将CP矩阵引入量子资源理论框架，其中CP态作为自由态，保CP信道作为自由操作。本文解决了他们工作中提出的几个问题。具体而言，我们给出了小维度下保CP信道的充要条件，这些条件在高维度下是必要的，并通过非负性的迹距离讨论了资源量化。通过构造一个明确的反例，我们证明了非负性的迹距离度量违反强单调性。我们还提供了另一种证明，即每个CPDNN信道$\Phi:\MM_n\to \MM_2$都是CPCP。此外，我们表明任何幺正CPDNN映射$\Phi:\MM_2\to \MM_n$也是CPCP。
英文摘要
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 $Φ:\MM_n\to \MM_2$ is CPCP. Additionally, we show that any unital CPDNN map $Φ:\MM_2\to \MM_n$ is also CPCP.
Comments13 pages (single column); comments/suggestions welcome!
