Sharpness Loss for Radon--Nikodym Derivatives of Completely Positive Maps
Abstract
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 ℰΛ,Ω. A sharp submap remains sharp if and only if ℰΛ,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.
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BibTeX
@misc{kian2026sharpness,
title = {{Sharpness Loss for Radon--Nikodym Derivatives of Completely Positive Maps}},
author = {Mohsen Kian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09915},
eprint = {2609.09915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09915},
}
RIS
TY - UNPB
TI - Sharpness Loss for Radon--Nikodym Derivatives of Completely Positive Maps
AU - Mohsen Kian
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09915
UR - https://arxiv.org/abs/2609.09915
AB - 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 ℰ_Λ,Ω. A sharp submap remains sharp if and only if 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.
ER -