Generalized Choi-Kraus dilations of linear maps between matrix algebras
Generalized Choi-Kraus dilations of linear maps between matrix algebras
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DOI：
10.7153/oam-2022-16-79
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发表时间：
2022
影响因子：
0.5
通讯作者：
Degu ng Han;Qianf ng Hu;Rui Liu
中科院分区：
文献类型：
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作者：
Degu ng Han;Qianf ng Hu;Rui Liu
.利用推广的Stinespring扩张定理，证明了两个矩阵代数Mn和Md之间的线性映射都具有n-同态扩张，因为这样的映射总是完全有界的.事实上，由于每个这样的映射都有一个广义Choi-Kraus表示<$（X）= ∑ Lk = 1 Ak XB <$k，它自动地通过表示矩阵系统{Ak，B}导出一个<$-同态膨胀，我们称之为线性映射的广义Choi-Kraus膨胀，当Ak = B时，称之为完全正（CP）映射的Choi-Kraus膨胀。本文的目的是研究广义Choi-Kraus扩张与其他已建立的扩张包括泛扩张之间的联系。证明了线性极小的广义Choi-Kraus扩张与线性极小的同态扩张等价，并给出了两个线性极小的广义Choi-Kraus扩张等价的充要条件. CP映射的线性极小Choi-Kraus伸缩是酉等价的，而线性极小广义Choi-Kraus伸缩，即使是CP映射，也不一定是等价的。事实上，线性映射只存在一个等价的线性极小广义Choi-Kraus扩张当且仅当其广义Choi矩阵满秩。
. By the generalized Stinespring’s dilation theorem, every linear map between two matrix algebras M n and M d has ∗ -homomorphism dilation due to the fact that such a map is always completely bounded. In fact, since every such a map has a generalized Choi-Kraus representation ϕ ( X ) = ∑ Lk = 1 A k XB ∗ k , it automatically induces a ∗ -homomorphism dilation by the representation matrix system { A k , B k } , which we call it the generalized Choi-Kraus dilation for a linear map, and the Choi-Kraus dilation when A k = B k for a completely positive (CP) map. The purpose of this paper is to examine the connections between the generalized Choi-Kraus dilations with other well-established dilations including the universal dilation. We prove that any linearly minimal ∗ -homomorphism dilation is equivalent to a linearly minimal generalized Choi-Kraus dilation, and present a necessary and suf fi cient condition for the equivalence of two linearly minimal generalized Choi-Kraus dilations. While all the linearly minimal Choi-Kraus dilations for a CP map are unitarily equivalent, the linearly minimal generalized Choi-Kraus dilations, even for a CP map, are not necessarily equivalent. In fact, a linear map admits only one equivalent class of linearly minimal generalized Choi-Kraus dilations if and only if its generalized Choi matrix has full rank.