一句话：用分配单体范畴证明量子信道范畴是纯量子操作范畴的典范完备化，并推广至拓扑丰富情形。
本文为量子信息与量子计算中纯态和混态之间的联系提出范畴论基础。作者采用分配单体范畴，证明全量子信道范畴可视为纯量子操作（含 ancilla 制备）范畴的典范完备化：有限维 C*-代数之间的完全正迹保持映射范畴，正是有限维向量空间与等距映射范畴的典范完备化。随后，作者将这一基础推广到拓扑丰富情形，以刻画量子信道之间的拓扑关系，并证明量子信道上的算子范数拓扑是由等距映射上的范数拓扑诱导的典范拓扑。
We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.
DOI 原文 ·
@inproceedings{paperbot5012,
title = {Quantum channels as a categorical completion},
author = {Mathieu Huot and Sam Staton},
booktitle = {LICS},
year = {2019},
doi = {10.1109/lics.2019.8785700}
}