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 epsilon(alpha)(i|jk) <= epsilon(alpha)(j|ik) + epsilon(alpha)(k|ij) holds for all subadditive bipartite entanglement measure epsilon, all permutations under parties i, j, k, all a. [0, 1], and all pure tripartite states. Then, we rigorously prove that the nonobtuse triangle area, enclosed by side epsilon(alpha) with 0 < alpha <= 1/2, is a measure for genuine tripartite entanglement. Finally, it is significantly strengthened for qubits that given a set of subadditive and nonsubadditive 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. Our results pave the way to study discrete and continuous multipartite entanglement within a unified framework.