Faithful geometric measures for genuine tripartite entanglement

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 Ei|jkα≤Ej|ikα+Ek|ijα holds for all subadditive bipartite entanglement measure E, all permutations under parties i,j,k, all α∈[0,1], and all pure tripartite states. Then, we rigorously prove that the nonobtuse triangle area, enclosed by side Eα with 0<α≤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 α>1, and the triangle area is not a measure for any α>1/2. Our results pave the way to study discrete and continuous multipartite entanglement within a unified framework.