完全正映射的Radon--Nikodym导数的尖锐性损失

中文摘要 AI 辅助

完全正映射的Radon--Nikodym定理将映射下方的序区间与其最小Stinespring表示换位中的正压缩对应起来。本文研究投影值Radon--Nikodym导数（尖锐子映射）在复合下的行为。我们证明尖锐性损失由诱导拉回映射的乘法缺陷决定。在有限维情形中，我们证明该缺陷由复合Kraus关系空间的正交补$\mathcal{E}_{\Lambda,\Omega}$决定。一个尖锐子映射保持尖锐当且仅当$\mathcal{E}_{\Lambda,\Omega}$约化提升的Radon--Nikodym投影。此外，我们建立了尖锐性全局保持的张量因子判据，并显式量化了尖锐性损失缺陷。

英文摘要

The Radon--Nikodym theorem for completely positive maps identifies the order interval below a map with positive contractions in the commutant of its minimal Stinespring representation. In this paper, we study the behavior of projection-valued Radon--Nikodym derivatives (sharp submaps) under composition. We prove that sharpness loss is determined by the multiplicative defect of the induced pullback map. In the finite-dimensional case, we show that this defect is determined by the orthogonal complement of the composite Kraus relation space, denoted $\mathcal{E}_{Λ,Ω}$. A sharp submap remains sharp if and only if $\mathcal{E}_{Λ,Ω}$ reduces the lifted Radon--Nikodym projection. Furthermore, we establish a tensor-factor criterion for the global preservation of sharpness and explicitly quantify the sharpness-loss defect.

发表机构

• University of Bojnord（博约德大学）

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