Faithful geometric measures for genuine tripartite entanglement

TL;DR

This paper introduces a geometric approach to quantify genuine tripartite entanglement using triangle relations and areas, applicable to various quantum systems, and establishes conditions under which these measures are valid.

Contribution

It provides a rigorous geometric interpretation of tripartite entanglement through triangle inequalities and areas, including bounds and generalizations for different quantum systems.

Findings

Triangle relation holds for all pure tripartite states and subadditive measures.

Non-obtuse triangle area with 0<α≤1/2 measures genuine tripartite entanglement.

Triangle area is not a valid measure for α>1/2 in qubit systems.

Abstract

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 $0<\alpha\leq 1/2$ is a measure for genuine tripartite entanglement. Useful lower and upper bounds for these measures are obtained, and generalizations of our results are also presented. Finally, it is significantly strengthened for qubits that, given a set of subadditive and non-additive measures, some state is always found to violate the triangle relation for any $\alpha>1$, 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.