Analytical proof of Gisin’s theorem for three qubits

Abstract

Gisin's theorem assures that for any pure bipartite entangled state, there is violation of Bell-CHSH inequality revealing its contradiction with local realistic model. Whether, similar result holds for three-qubit pure entangled states, remained unresolved. We show analytically that all three-qubit pure entangled states violate a Bell-type inequality, derived on the basis of local realism, by exploiting the Hardy's non-locality argument.

Identification of three-qubit entanglement

“…An important approach to characterize entanglement is Bell inequality [3][4][5][6][7][8]. For instance, N. Gisin has proved that all two-qubit pure entangled states violate the CHSH inequality [4].…”

Generalized Hardy’s Paradox

“…Hardy's paradox, with post-selections taken into consideration, therefore stands out among the others, since (i) it applies to the two-party scenario; (ii) it can be generalized to multi-party and high-dimensional scenar-ios [14] (hereafter we would like to call Cereceda's version of n-qubit Hardy's paradox/inequality as the standard Hardy's paradox/inequality, to distinguish them from the most general ones that we shall present in this paper); and (iii) inequalities constructed based on it allow to detect more entangled states and provide a key element to prove Gisin's theorem [31,32] -which states that any entangled pure state violates Bell's inequality [33]. The GHZ paradox does not share most of these merits (see also the Mermin-Ardehali-Belinskii-Klyshko inequality [34][35][36], which was also a kind of generalization of CHSH inequality to n qubits, but was not violated by all pure entangled states, even not by all the generalized GHZ states).…”

Bell inequalities for multipartite qubit quantum systems and their maximal violation

“…(20) we have used the fact that the maximum of x cos θ + y sin θ taking over all θ is x 2 + y 2 . Formulas (14) and (15) can be similarly proven.…”

“…Due to their significance, Bell inequalities have been generalized from the two-qubit case, such as the Clauser-Horne-Shimony-Holt (CHSH) inequality [10] to the N -qubit case, such as the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality [11,12], and to arbitrary d-dimensional (qudit) systems such as the Collins-Gisin-Linden-Masser-Popescu inequality [13]. However, except for some special cases such as bipartite pure states [2,3,6], three-qubit pure states [5,14], and general two-qubit quantum states [15], there are no Bell inequalities yet that can be violated by all the entangled quantum states, although it is shown recently that any entangled multipartite pure states should violate a Bell inequality [7]. Thus it is of great importance to find more effective Bell-type inequalities to detect the quantum entanglement.…”