量子信道的乘法性质
Multiplicative Properties of Quantum Channels
摘要：
本文研究了矩阵代数上保迹完全正映射所描述的量子信道的乘法行为。结果表明，单位量子信道的乘法域与其光谱性质有着密切的联系。给出了一个结构定理，揭示了子代数上任意单位量子信道的自守性质。各种类型的量子信道（不可约的、原始的等）然后分析在这个稳定子代数。引入了单位量子信道的乘性指数的概念，它度量了单位信道为了使乘性代数稳定而需要与自身合成的次数。我们证明了在所有酉完全正映射的集合中，具有平凡乘法域的映射在完全有界范数拓扑中是稠密的。讨论了它在量子信息理论中的一些应用。
英文摘要：
In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem, which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive etc.) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed.
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