凸锥的纯化
Purifications for Convex Cones
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中文总结 AI 辅助
该研究利用有限维真凸锥的几何，证明不可分解齐次锥内点的纯化存在性，给出局部自同构下的唯一性判据，并通过多种锥的例子说明相关现象。
中文摘要 AI 辅助
受量子理论和广义概率理论中纯化原理重要性的启发，我们仅利用有限维真凸锥的几何来研究纯化。我们证明了关于不可分解锥和包含最大纠缠态的中间张量锥的存在性定理；特别地，每个不可分解齐次锥的内点都存在纯化，例如洛伦兹锥。我们还给出了在局部自同构下的唯一性判据。在边界上，我们证明若C的每个真面都是单纯形，则仅纯点可存在纯化，并说明存在非单纯形面时该结论不成立。涉及半正定锥、洛伦兹锥、k正映射、PPT张量和多面体锥的例子，展示了存在性与非唯一性现象。
英文摘要
Motivated by the importance of the purification principle in quantum theory and generalized probabilistic theories, we study purifications using only the geometry of a finite-dimensional proper convex cone. We prove an existence theorem for indecomposable cones and intermediate tensor cones containing the maximally entangled state; in particular, every interior point of an indecomposable homogeneous cone admits a purification. This applies to Lorentz cones, for example. We also give a criterion for uniqueness up to local automorphisms. On the boundary, we show that if every proper face of $C$ is simplicial, then only pure points can admit purifications, and we demonstrate that this conclusion fails in the presence of non-simplicial faces. Examples involving positive semidefinite cones, Lorentz cones, $k$-positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.
