Tripartite Entanglement versus Tripartite Nonlocality in Three-Qubit Greenberger-Horne-Zeilinger-Class States

Abstract

We analyze the relationship between tripartite entanglement and genuine tripartite nonlocality for three-qubit pure states in the Greenberger-Horne-Zeilinger class. We consider a family of states known as the generalized Greenberger-Horne-Zeilinger states and derive an analytical expression relating the three-tangle, which quantifies tripartite entanglement, to the Svetlichny inequality, which is a Bell-type inequality that is violated only when all three qubits are nonlocally correlated. We show that states with three-tangle less than 1/2 do not violate the Svetlichny inequality. On the other hand, a set of states known as the maximal slice states does violate the Svetlichny inequality, and exactly analogous to the two-qubit case, the amount of violation is directly related to the degree of tripartite entanglement. We discuss further interesting properties of the generalized Greenberger-Horne-Zeilinger and maximal slice states.

Theory of Genuine Tripartite Nonlocality of Gaussian States

“…In recent years, nonlocality has come to be appreciated as an operational resource [9], which plays an essential role, e.g., for the implementation of secure quantum key distribution [10]. The concept of multipartite nonlocality is, however, significantly less understood than its bipartite counterpart [3][4][5]9,[11][12][13][14][15][16][17]. For a tripartite system, the conventionally employed inequality, whose violation signals a genuine three-way nonlocality, is due to Svetlichny [12].…”

“…which can be instead violated quantum mechanically up to the value jSj ¼ 4 ffiffi ffi 2 p ≈ 5.65, reached for instance on threequbit GHZ states [12,14,15]. Any violation of inequality (4) is a signature of genuine tripartite nonlocality on the state ρ, and implies that ρ is fully inseparable.…”

“…According to this framework, it has been conjectured that all fully inseparable tripartite pure states are three-way nonlocal [17]. In general, the interplay between the quantitative violation of these inequalities and the degree of genuine tripartite entanglement in the tested states remains unclear [14]. We recall that multipartite entanglement, i.e., entanglement encoded in systems of more than two parties, can arise in many inequivalent forms [6].…”

“…For a tripartite system, the conventionally employed inequality, whose violation signals a genuine three-way nonlocality, is due to Svetlichny [12]. Such an inequality can be violated, e.g., by both Greenberger-Horne-Zeilinger (GHZ) and W classes of states for three qubits [14,15], and is essentially the unique witness of nonlocality when all three parties perform two measurements with two outcomes each [17]. More recently, a reassessment of tripartite nonlocality has led to a series of weaker inequalities, whose violation can reveal tripartite nonlocality even when the Svetlichny one is not violated [9,17].…”

Detecting fullN-particle entanglement in arbitrarily-high-dimensional systems with Bell-type inequalities

“…Given N -particle quantum systems, the correlations among them have been the subject of several recent studies [3][4][5][6][7][8]. This is also motivated by the question whether the correlations in recent experiments on 3-or 4-particle systems are due to the full N -particle entanglement and not just combinations of quantum entanglement of smaller number of particles [9,10].…”