Faithful geometric measures for genuine tripartite entanglement*

We present a faithful geometric picture for genuine tripartite entanglement of discrete, continuous, and hybrid quantum systems. We first find that the triangle relation $mathcal{E}^alpha_{i|jk}leq mathcal{E}^alpha_{j|ik}+mathcal{E}^alpha_{k|ij}$ holds for all subadditive bipartite entanglement measure $mathcal{E}$, all permutations under parties $i, j, k$, all $alpha in [0, 1]$, and all pure tripartite states. It provides a geometric interpretation that bipartition entanglement, measured by $mathcal{E}^alpha$, corresponds to the side of a triangle, of which the area with $alpha in (0, 1)$ is nonzero if and only if the underlying state is genuinely entangled. Then, we rigorously prove the non-obtuse triangle area with $01$, and the triangle area is not a measure for any $alpha>1/2$. Hence, our results are expected to aid significant progress in studying both discrete and continuous multipartite entanglement.