发表机构
Faculty of Mathematics, Physics and Informatics, University of Gdańsk（格但斯克大学数学、物理与信息学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对III型冯·诺依曼代数系统，重新表述了Choi--Jamiołkowski同构，证明其成立的条件，明确了正规映射空间及完全正性的相关性质，填补了无穷维量子信息理论中该对应关系的空白。
AI 中文摘要
Choi--Jamiołkowski同构将完全正映射与双体态对应起来，是有限维量子信息理论的重要基础。对于由III型冯·诺依曼代数建模的具有无穷多自由度的系统，既没有迹也没有密度矩阵可用，因此必须重新表述该同构。我们证明，对于任意冯·诺依曼代数$\boldsymbol{\textit{M}}$和$\boldsymbol{\textit{N}}$，从$\boldsymbol{\textit{M}}$到$\boldsymbol{\textit{N}}$预对偶的正规完全有界映射空间，与$\boldsymbol{\textit{M}}$和$\boldsymbol{\textit{N}}$的反代数的空间张量积的预对偶之间，存在一个典范序同构；在此对应下，完全正性对应正性。若要将反代数替换为$\boldsymbol{\textit{N}}$本身，则需要$\boldsymbol{\textit{N}}$与其自身存在反同构，且我们证明，只要要求该对应在$\boldsymbol{\textit{M}}$上是自然的，该条件不仅充分且必要。这类要求是不可避免的，因为对于每个自反同构的$\boldsymbol{\textit{M}}$，该同构会因平凡原因存在。因此，Connes构造的III型因子不存在Choi--Jamiołkowski对应。在此过程中，我们还证明，所有正规映射的空间（赋予算子范数）对于此目的过大，且在该语境下完全正性不强制完全有界性。
英文摘要
Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on $\mathcal{M}\bar{\otimes}\mathcal{N}$ but on the tensor product of $\mathcal{M}$ with the \emph{opposite} algebra of $\mathcal{N}$, or equivalently with a commutant; returning to $\mathcal{N}$ itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is $*$-anti-isomorphic to itself. Naturality cannot be dropped: whenever $\mathcal{M}$ is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.
Comments19 pages
