Knowledge graph embedding (KGE) models typically represent each relation as an operator on entity embeddings. In this work, we identify three structural axioms that any principled relation operator must satisfy, linearity, trace preservation, and complete positivity, and show that they characterize a Kraus channel structure via the Kraus representation theorem. The completeness constraint defining this family is equivalent to these axioms, providing a principled foundation rather than an externally imposed condition. Under this formulation, most existing operator-based KGE models are recoverable as special cases with Kraus rank $κ= 1$ under specific embedding choices. We further generalize this characterization to arbitrary metric geometries by introducing \mbox{w-Kraus} channels, which satisfy completeness by construction within their respective spaces. Building on this theory, we propose \textsc{KrausKGE}, a principled KGE model that naturally handles $1$-to-$N$ and $N$-to-$N$ relations, supports $k$-hop reasoning without requiring explicit path encoders, and eliminates the need for norm constraints on entity embeddings. Additionally, our framework yields the first theoretically grounded per-relation complexity measure in the KGE literature, with a provable lower bound in terms of the empirical relation matrix rank. Empirical evaluation demonstrates that \textsc{KrausKGE} consistently outperforms strong baselines on $N$-to-$N$ relations, with performance gains that increase monotonically with relation fan-out, in alignment with theoretical predictions.

翻译：知识图谱嵌入（KGE）模型通常将每个关系表示为作用于实体嵌入的算子。本文识别出任何合理的关系算子必须满足的三个结构公理——线性性、迹保持性和完全正性，并借助克劳斯表示定理证明这些公理刻画了克劳斯通道结构。定义该族算子的完全性约束与这些公理等价，从而提供了理论上的基础而非外部强加的条件。在此框架下，现有的大多数基于算子的KGE模型均可恢复为克劳斯秩$κ=1$且采用特定嵌入选择的特例。我们进一步通过引入\mbox{w-Kraus}通道将此刻画推广至任意度量几何空间，该通道在其各自空间内通过构造满足完全性性质。基于该理论，我们提出\textsc{KrausKGE}这一具有理论基础的KGE模型：其自然处理$1$-对-$N$和$N$-对-$N$关系，支持无需显式路径编码器的$k$跳推理，并消除对实体嵌入的范数约束需求。此外，本文框架首次给出KGE文献中具有理论基础的逐关系复杂度度量，并提供了基于经验关系矩阵秩的可证明下界。实验评估表明，\textsc{KrausKGE}在$N$-对-$N$关系上持续优于强基线模型，其性能增益随关系扇出度单调递增，与理论预测一致。